Cofinality and Measurability of the First Three Uncountable Cardinals

COFINALITY AND MEASURABILITY OF THE FIRST THREE UNCOUNTABLE CARDINALS

ARTHUR W. APTER, STEPHEN C. JACKSON, BENEDIKT LOWE¨

Abstract. This paper discusses models of set theory without the Axiom of Choice. We investigate all possible patterns of the cofinality function and the distribution of measurability on the first three uncountable cardinals. The result relies heavily on a strengthening of an unpublished result of Kechris: we prove (under AD) that there is a cardinal κ such that the triple (κ, κ+, κ++) satisfies the strong polarized partition property.

1. Introduction

In ZFC, small cardinals such as ℵ1, ℵ2, and ℵ3 cannot be measurable, as measurability implies strong inaccessibility; they cannot be singular either, as successor cardinals are always regular. So, in ZFC, these three cardinals are non-measurable regular cardinals. But both of the mentioned results use the Axiom of Choice, and there are many known situations in set theory where these small cardinals are either singular or measurable: in the Feferman-Lévymodel, ℵ1 has countable cofinality (cf. [Jec03, Example 15.57]), in the model constructed independently by Jech and Takeuti, ℵ1 is measurable (cf. [Jec03, Theorem 21.16]), and in models of AD, both ℵ1 and ℵ2 are measurable and cf(ℵ3) = ℵ2 (cf. [Kan94, Theorem 28.2, Theorem 28.6, and Corollary 28.8]). Simple adaptations of the Feferman-Lévyand Jech and Takeuti arguments show that one can also make ℵ2 or ℵ3 singular or measurable, but is it possible to control these properties simultaneously for the three cardinals ℵ1, ℵ2 and ℵ3? In this paper, we investigate all possible patterns of measurability and cofinality for the three mentioned cardinals. Combinatorially, there are exactly 60 (= 3 × 4 × 5) such patterns: ℵ1 can be measurable, regular non-measurable, or singular (3 possibilities); ℵ2 can be measurable, regular non-measurable, or have cofinality ℵ1 or ℵ0 (4 possibilities); and ℵ3 can be measurable, regular non-measurable, or have cofinality ℵ2, ℵ1, or ℵ0 (5 possibilities). Out of these 60 patterns, 13 are impossible for trivial reasons: for instance, if ℵ1 is singular, then ℵ2 cannot have cofinality ℵ1, as cofinalities always must be regular cardinals.

Date: May 4, 2009. 2000 Mathematics Subject Classification. Primary 03E02, 03E35, 03E55, 03E60; secondary 03E10, 03E15, 03E25. The first author’s visits to Amsterdam and the third author’s visit to New York were partially supported by the DFG-NWO Bilateral Grant KO 1353-5/1/DN 62-630. In addition, the first author wishes to acknowledge the support of various PSC-CUNY and CUNY Collaborative Incentive grants. The third author would like to thank the CUNY Graduate Center Mathematics Program for their hospitality and partial financial support during his stay in New York. 1 For these patterns, we shall use the labels M and ℵn, standing for “measurable” and “non-measurable and cofinality ℵn”, respectively, and write

[ x1 / x2 / x3 ] for the statement “ℵ1 has property x1, ℵ2 has property x2, and ℵ3 has property x3”. For technical reasons, we shall assume that all measurable cardinals carry a normal ultrafilter. In all of our constructions of models, this will be the case; this slightly non-standard definition does make a difference for lower bounds. We discuss this in somewhat more detail in §9. We shall go through all 60 possible patterns and prove 47 of them to be consistent from the appropriate large cardinal axioms, and 13 of them to be inconsistent. Instead of proving a ridiculously large number of results, we have arranged the paper as follows: in § 2 we shall provide some basic tools for forcing in the ZF context, some of them using polarized partition properties. These tools will allow us to look at the 47 consistent patterns as a graph reducing all cases to eight base cases in § 3. In §§ 4 and 5, we prove the consistency of all base cases. In the former section, we use techniques from forcing and large cardinals; in the latter, we rely on AD (and in particular, on certain polarized partition properties that hold under AD). The main polarized partition property we use is a generalization of an unpublished theorem of Kechris from the 1980s (cf. [AH86, p. 600]). In this paper, we give a proof of (a slightly stronger version of) Kechris’ result (§ 6) and generalize it to higher exponents (§§ 7 and 8), as needed in our applications in § 5. Finally, in § 9, we shall summarize upper and lower consistency strength bounds of all 60 patterns.

2. A Toolkit In this section, we list a number of basic forcing facts, the main deﬁnitions of polarized partition properties and some facts about AD on which our analysis rests. 2.1. Forcing facts.

Theorem 1. If V |= ZF+“κ is a measurable cardinal”, Pκ is Pˇr´ıkrýforcing for κ, and G is Pκ-generic over V, then in the generic extension V[G], cf(κ) = ℵ0, any cardinal having cofinality κ in V now has cofinality ℵ0, and the cofinalities and measurability of all other cardinals are unchanged.

Proof. This is [Apt96, Lemmas 1.2, 1.3 and 1.5]. q.e.d.

Theorem 2. If V |= ZF+“ℵ1 is measurable”, Add(ω, ω1) is the partial order for adding ω1 many Cohen reals, and G is Add(ω, ω1)-generic over V, then in the generic extension V[G], ℵ1 is regular but non-measurable and coﬁnalities and measurability of all other cardinals are unchanged.

Proof. The fact that ℵ1 becomes non-measurable is a special case of the general ZF-result (due to Ulam) that if κ injects into 2λ for some λ < κ, then there cannot be a κ-complete ultraﬁlter on κ (cf. [Kan94, Theorem 2.8]). Obviously, the ω1- ω sequence of Cohen reals produces an injection of ω1 into 2 . Since Add(ω, ω1) is canonically well-orderable and |Add(ω, ω1)| = ℵ1, the proof that all cardinals and 2 coﬁnalities are preserved is the same as when AC is true. Since |Add(ω, ω1)| = ℵ1, the argument given in the proof of [AH86, Lemma 2.1] shows that the measurability of all cardinals greater than ℵ1 is preserved. q.e.d.

Theorem 3. If V |= ZFC+“κ < λ are measurable cardinals”, then for x3 ∈

{ℵ0, ℵ1, ℵ2, ℵ3, M}, there is a symmetric submodel Nx3 satisfying [ M / ℵ2 / x3 ]. If x3 6= M, only one measurable cardinal is needed in V. Proof. We sketch the proof of Theorem 3. Without loss of generality, we assume that GCH holds in V. Let G0 be Col(ω, <κ)-generic over V, where for ρ < ζ, ρ a regular cardinal, ζ a cardinal, Col(ρ, <ζ) is the L´evycollapse of all cardinals less than ζ to ρ. For H which is Col(ρ, <ζ)-generic over V and ξ ∈ (ρ, ζ) a cardinal, let + H¹ξ be all elements of H which are members of Col(ρ, <ξ). Let G1 be Col(κ , <γ)- generic over V, where γ is either κ+ω, κ+κ, κ+η for η = κ+, or λ. We write HDV(X) for the class of sets hereditarily V-deﬁnable with a parameter from X.

Consider the symmetric model Nx3 := HDV({G0¹δ ; δ ∈ (ω, κ) and δ is a cardinal}∪ + + {G1¹δ ; δ ∈ (κ , γ) and δ is a cardinal}). Since in V, there is a κ sequence of subsets of κ, standard arguments show that Nx3 is a model for [ M / ℵ2 / ℵ0 ], +ω +κ +η [ M / ℵ2 / ℵ1 ], [ M / ℵ2 / ℵ2 ], or [ M / ℵ2 / M ], for γ either κ , κ , κ for η = κ+, or λ respectively. Since in V, there is a κ+ sequence of subsets of κ and a κ++ + sequence of subsets of κ , Nx3 := HDV({G0¹δ ; δ ∈ (ω, κ) and δ is a cardinal}) is a model for [ M / ℵ2 / ℵ3 ]. Clearly, the only time a second measurable cardinal is needed in the construction is for the pattern [ M / ℵ2 / M ]. q.e.d.

Theorem 4. Suppose i ∈ ω. Let V |= ZF + “κ is a limit cardinal” + “λ := κ+i”. Let G be Col(ω, <κ)-generic over V. Consider the model M obtained by symmetrically collapsing κ to ℵ1, i.e., the model M := HDV({G¹δ ; δ ∈ (ω, κ) and δ is a cardinal}). Then the following hold:

(i) If V |= “λ is measurable”, then M |=“λ = ℵi+1 is measurable”. +j (ii) If V |= “cf(λ) = κ for some j ≤ i”, then M |= “cf(λ) = ℵj+1”. Proof. Since V |= “Col(ω, <κ) is canonically well-orderable and |Col(ω, <κ)| = κ”, (i) follows from the argument given in the proof of [AH86, Lemma 2.1], the fact that κ = ℵ1 in M, and the fact that cardinals at and above κ are preserved to M; (ii) follows from the fact that κ = ℵ1 in M and the fact that cardinals and coﬁnalities at and above κ are preserved to M. q.e.d.

2.2. Polarized partition properties. Fix a strictly increasing triple (κ0, κ1, κ2) of cardinals and an ordinal δ ≤ κ0. We say a function f : 3 × δ → On is a block function if κi−1 < f(i, α) < κi for i ∈ 3 (and κ−1 := 0), and we say it is increasing if f(i, α) < f(i, β) whenever α < β. We denote the set of increasing block functions by IBFδ. If H~ = (H0,H1,H2) is a tuple such that Hi ⊆ κi (for i ∈ 3), we deﬁne a subset FH~ ⊆ IBFδ by

f ∈ FH,δ~ : ⇐⇒ for all α ∈ δ and i ∈ 3, we have f(i, α) ∈ Hi.

If P ⊆ IBFδ is a partition of all increasing block functions into two disjoint sets, ~ we call a triple H δ-homogeneous for P if either FH,δ~ ⊆ P or FH,δ~ ∩ P = ∅. 3 S For Ramsey-type partition properties, we also define the set IBF<δ := α<δ IBFα, and for a partition P ⊆ IBF<δ, we say that a tuple H~ is <δ-homogeneous for P if for all α < δ, either FH,α~ ⊆ P or FH,α~ ∩ P = ∅. Definition 5. The polarized partition property δ (κ0, κ1, κ2) → (κ0, κ1, κ2) is the statement that for every partition P , there is a δ-homogeneous tuple H~ with 1 |Hi| = κi. If δ = κ0, we call it the strong polarized partition property. The Ramsey-type polarized partition property <δ (κ0, κ1, κ2) → (κ0, κ1, κ2) is the statement that for every partition P , there is a <δ-homogeneous tuple H~ with |Hi| = κi. Polarized partition properties with pairs of cardinals instead of triples are defined analogously.2

κ0 Lemma 6. Suppose that (κ0, κ1, κ2) → (κ0, κ1, κ2) and κ0 > ℵ1. Then the following hold:

κ0 (i) (κ1, κ2) → (κ1, κ2) , and <ω1 (ii) (κ1, κ2) → (κ1, κ2) . Proof. Claim (i) is trivial. Claim (ii) follows by the standard methods developed by Kleinberg for the standard partition relations [Kle70, Lemmas 1.3 and 1.4] that easily transfer to the polarized case (as mentioned in [AHJ00, Facts 4.3 through κ0 ω1+ω1 4.7]): (κ1, κ2) → (κ1, κ2) implies (κ1, κ2) → (κ1, κ2) , from which we get ω1 ℵ1 (κ1, κ2) → (κ1, κ2)2ω1 , the partition relation for partitions into 2 many sets. <ω1 From this, we get (κ1, κ2) → (κ1, κ2) by coding. q.e.d.

As is the case for ordinary partition relations, the polarized partition property is equivalent to a c.u.b. version. A block function f is said to be of uniform coﬁnal- ity ω if there is a function g : 3×δ ×ω → On such that f(i, α) = sup{g(i, α, n); n ∈ ω}, and g is strictly increasing in the last argument. We say that a block function f is of the correct type if it is increasing, everywhere discontinuous, and of uniform coﬁnality ω. We write CTFδ for the functions of the correct type. If P ⊆ CTFδ, we ~ call a triple H δ-c.u.b.-homogeneous if either FH,δ~ ∩CTFδ ⊆ P or FH,δ~ ∩P = ∅.

c.u.b. δ Deﬁnition 7. We say (κ0, κ1, κ2) −→ (κ0, κ1, κ2) if for every partition P ⊆ CTFδ, there is a triple C~ = (C0,C1,C2) such that Ci is a closed unbounded set in κi (for i ∈ 3) and C~ is δ-c.u.b.-homogeneous.

c.u.b. δ Fact 8. For any δ ≤ κ0 < κ1 < κ2, we have that (κ0, κ1, κ2) −→ (κ0, κ1, κ1) δ ω·δ implies (κ0, κ1, κ2) → (κ0, κ1, κ1) . Also, (κ0, κ1, κ2) → (κ0, κ1, κ1) implies c.u.b. δ (κ0, κ1, κ2) −→ (κ0, κ1, κ1) .

Proof. The easy argument can be found in [Jac08, Lemma 3.3]. q.e.d.

1Note that this terminology differs from that of [AHJ00]. 2Note that as in [AHJ00, Definition 4.11], these definitions are equivalent to the partition theoretic ones found in [AHJ00, Definition 4.1sq]. 4 2.3. Magidor-like forcing. In [Hen83], Henle introduced Magidor-like forcing for controlling the cofinalities of cardinals in choiceless contexts in the presence of partition properties. Assuming that κ → (κ)<δ and that δ is a regular, uncountable cardinal, Magidor-like forcing changes the cofinality of κ to δ without adding any bounded subsets to κ (thereby preserving the fact that κ is a cardinal; cf. [Hen83, Proposition 1.3]). We define the set Pδ,κ by [ \ <δ κ Pδ,κ = {hs, xi ; s ∈ [κ] , x ∈ [κ] , s < x}. κ S We use hxi to denote {ωq ; q ∈ [x] }, where ωq = { n<ω q(α + n); α < κ}. 0 0 The partial ordering for Pδ,κ is now defined by saying that hs , x i extends hs, xi 0 0 0 <δ if and only if s ⊆ s , hx i ⊆ hxi, and s \ s = ωt for some t ∈ [x] . For p ∈ Pδ,κ, we denote the coordinates of p by p0 and p1, i.e., p = hp0, p1i. This was generalized in [AHJ00, §6] to the context of polarized partition properties. In the following, we shall need two preservation results from [AHJ00]:

κ0 Lemma 9 (Initial segment preservation). If (κ0, κ1, κ2) → (κ0, κ1, κ2) , then after κ0 forcing with Pκ0,κ2 , we still have (κ0, κ1) → (κ0, κ1) .

Proof. This follows from the proof of [AHJ00, Proposition 6.3]. q.e.d.

<ω1 Lemma 10 (Countable ﬁnal segment preservation). If (κ0, κ1) → (κ0, κ1) and <ω1 γ < κ0 is regular, then after forcing with Pγ,κ0 , the relation κ1 → (κ1) remains true.

Proof. This follows from the proof of [AHJ00, Proposition 6.4]. q.e.d.

2.4. The Axiom of Determinacy and Suslin Cardinals. Let us recall some basic deﬁnitions from descriptive set theory. By a boldface pointclass Γ we mean a collection of sets of reals closed under continuous preimages. For a pointclass Γ we let Γ˘ denote the dual pointclass Γ˘ := {A ; ωω\A ∈ Γ}. A pointclass Γ is called selfdual if Γ = Γ˘, and non-selfdual otherwise. If Γ is non-selfdual, we can deﬁne ∆ := Γ ∩ Γ˘. We say that a non-selfdual pointclass Γ has the separation property (in symbols: Sep(Γ)) if any two disjoint sets in Γ can be separated by a set in ∆. In general, at most one of Γ and Γ˘ can have the separation property. In [Ste81b], Steel proved that AD implies that one of the two does. From now on in this section, we shall assume AD. The class of pairs of non-selfdual pointclasses (Γ, Γ˘) (i.e., Γ 6= Γ˘) such that one ω of them is closed under ∃ω has order type Θ := sup{α; there is a surjection from ωω onto α}.

We call these the L´evy pointclasses. Let (Γα, Γ˘ α) be the αth such pair. If one of ωω 1 them is not closed under ∀ , the other one is. If this is the case, we let Σα be the ωω 1 one that isn’t. If both of them are closed under ∀ , let Σα be the one with the 1 1 separation property. As usual, Πα := (Σα)˘. In [KSS81, §4], the authors proposed a classiﬁcation of these pointclasses. They fall into four types of “projective like hierarchies” which are distinguished by the closure properties of the pointclass at the base of the hierarchy (this is recalled after Proposition 13 below). For example, 1 1 if cf(α) = ω, then Σα is at the base of a type I hierarchy. In this case Σα is the 5 collection of sets which can written as a countable union of sets each of which is in 1 Σβ for some β < α. These pointclasses play a particularly important role in the arguments of §§6–8. As usual, a set of reals A is called λ-Suslin if there is a tree T ⊆ (ω × λ)<ω such that A = p[T ] := {x ; ∃y ∈ λω((x, y) ∈ [T ])}. We write S(λ) for the pointclass of ω all λ-Suslin sets. These pointclasses are closed under ∃ω , and thus show up in our list mentionedS in the last paragraph. A cardinal κ is called a Suslin cardinal if S(κ)\ λ<κ S(λ) 6= ∅. We give the well-known Kunen-Martin theorem (cf. [Kec78, Theorem 3.11]) with its proof, as the general idea of this proof will be used repeatedly in our results of §§6–8. Theorem 11 (Kunen-Martin). Let ≺ be a κ-Suslin wellfounded relation on ωω. Then the rank of ≺ is less than κ+.

Proof. Let T be a tree on ω × ω × κ with ≺ = p[T ]. Let U be the wellfounded tree consisting of ﬁnite ≺-decreasing sequences (x0, . . . , xn), that is, xn ≺ · · · ≺ x1 ≺ x0. It is easy to see that ≺ and U have the same rank. To each ~x = (x0, . . . , xn) ∈ U assign π(~x) = (x0¹n + 1, . . . , xn¹n + 1, `(x1, x0)¹n + 1, . . . `(xn, xn−1)¹n + 1), where ω `(y, z) ∈ κ is the leftmost branch of Ty,z. If ~y extends ~x, we view π(~y) as extending π(~x) in a natural way. The map π is order-preserving from U into a wellfounded relation on a set in bijection with κ. Thus the rank of ≺ must be less than κ+. q.e.d.

In [Ste83, Theorem 4.3], Steel identiﬁes (assuming V=L(R)) the pointclasses 1 1 S(κ) in the list of Σαs and Παs and calculates their Suslin cardinals. For instance, R 3 R 1 if κ is the least non-hyperprojective ordinal , we have S(κ ) = ΠκR = IND, and κR is a Suslin cardinal as witnessed by the inductive sets. Proposition 12. If AD holds, then there are weakly inaccessible Suslin cardinals.

Proof. As just mentioned, Steel’s analysis of scales in L(R) shows that κR is a Suslin cardinal. In [KKMW81, Theorem 3.1], the authors show that it is in fact weakly Mahlo. Note that κR is by no means the only (or smallest) weakly inaccessible Suslin cardinal (cf. [Ste81a, Theorem 3.1]). q.e.d.

By the work of [KKMW81] mentioned in the proof of Proposition 12, the analysis of the scale property of pointclasses is closely connected to partition properties. Our results from §7 and §8 can be seen as an extension of this work. In the following overview, we follow [Jac08, p. 295–297]: For a selfdual pointclass ∆, we let o(∆) := sup{|A|W ; A ∈ Γ} and δ(∆) := sup{α ; there is a ∆-prewellordering of length α}. Note that (under AD) if ∆ is ω closed under ∃ω and finite intersections, then o(∆) = δ(∆) [KSS81, Theorem 2.3.1]. Let us fix the increasing enumeration of all Suslin cardinals hκα ; α < Ξi. Note that ZF + AD does not fix the value of Ξ: the Suslin cardinals could be unbounded below Θ (in this case, every set has a scale and thus by a result of

3Here, IND is the pointclass of inductive sets, and the pointclass HYP := IND ∩ IND˘ is ω ω the class of hyperprojective sets. All three mentioned pointclasses are closed under ∃ω and ∀ω , and we have Sep(IND˘). 6 Woodin [Kan94, Theorem 32.23], ADR holds) or there could be a largest Suslin cardinal. It is enough for the results of this paper to consider the Suslin cardinals 2 L(R) in L(R), i.e., hκα ; α < (δ1) i. Here [Ste83] gives a complete analysis of the Suslin cardinals. We note though that the main partition result we prove in §8 only uses AD. We recall some facts about the classification of projective-like hierarchies and how this pertains to Suslin cardinals. The facts we review below are sufficient for the results of this paper. The reader can consult [Ste81a] and [Ste83] for more details. We note that the latter paper assumes V=L(R). Although we don’t need it for the results of this paper, [Jac09] presents the theory of the Suslin cardinals from just AD. Following [Ste81a] consider ω D := {o(∆); ∆ is selfdual and closed under ∧, ∃ω }. Clearly, D consists of limit ordinals and is closed unbounded in Θ. Each α ∈ D corresponds to the base of a projective-like hierarchy. If cf(α) = ω this is called a type I hierarchy. In this case the Wadge degree of rank α is selfdual and consists α of a countable join of sets of lower Wadge rank. We let Σ0 in this case be the α collection of countable unions of sets of Wadge rank below α. We let Π0 be the α α dual class, and define Σn, Πn for n > 0 as usual. This defines the projective-like α α hierarchy. In this case, Σ0 , Π1 , etc. have the prewellordering property. These classes will be particularly important for the arguments of §§6–8. If cf(α) > ω, the Wadge pair (Γ, Γ˘) of rank α is non-selfdual. By [Ste81a], exactly one of Γ, Γ˘, say ω Γ˘, has the separation property, and this class is closed under ∃ω . If this pointclass ω is not also closed under ∀ω we are in type II if Γ is not closed under ∨ and in type III if Γ is closed under ∨. We call Γ in these cases the Steel pointclass at the base of the hierarchy. If Γ is closed under real quantification then we are in type IV (in this case the projective-like hierarchy is built up by applying quantifiers to Γ ∧ Γ˘). This analysis of the projective-like hierarchies does not depend on the scale property, and assumes just AD. We now specialize to the Suslin cardinals and Suslin pointclasses. We say the λth Suslin cardinal κλ is a limit Suslin cardinal if λ is a limit ordinal, and otherwise a successor Suslin cardinal (so a successor Suslin cardinal may be a limit cardinal). First we recall that [Ste83] shows that the Suslin cardinals form a closed set in L(R) 2 L(R) (with largest element (δ1) ). More generally, Steel and Woodin have shown that the Suslin cardinals are closed below Θ assuming AD+, and closed below their supremum assuming just AD. So we have:

Proposition 13. If λ is a limit ordinal, then κλ is a limit of Suslin cardinals. S If κ = κλ is a limit Suslin cardinal, then ∆ := ρ<κ S(ρ) is selfdual and closed ω under ∧, ∃ω , and so o(∆) ∈ D. As discussed above, ∆ sits at the base of a projective-like hierarchy in one of four possible types. In [Ste83], Steel identifies 1 1 the pointclasses S(κ) among the Σα and Πα and in fact determines the scaled Lévy 1 1 classes among the Σα, Πα (assuming V=L(R)). We recall some of the consequences in terms of the possible hierarchy types. In all cases, κ = o(∆) = δ(∆). α Type I: In this case cf(κ) = cf(λ) = ω. If we let, as above, Σ0 (where α = o(∆) = κ) be the collection of countable unions of sets in ∆, then α α + α Σ0 , Π1 , etc. have the scale property. κ is a Suslin cardinal, and a Π1 7 α + α scale on a Π1 -complete set has norms of length κ . We have S(κ) = Σ1 + α and S(κ ) = Σ2 . Type II or III: In this case, let Γ be the Steel class defined above. So, ω ω ∆ = Γ ∩ Γ˘, and Γ is closed under ∧, ∀ω . Then S(κ) = ∃ω Γ, and Scale(Γ), Scale(S(κ)) hold. Type IV: In this case, the pointclasses Γ, Γ˘ of Wadge degree κ are closed under real quantification. Let Γ be such that Γ˘ has the separation property. Then Scale(Γ), and S(κ) = Γ. If κ is a regular limit Suslin cardinal, then [Ste81a, Theorem 2.1] shows that Γ (as in the above hierarchy descriptions) is closed under ∨. Thus, we are in type III or IV. Finally, the analysis of [Ste83, Theorem 4.3] shows that a successor Suslin cardinal is either a successor cardinal or has cofinality ω. Thus, a weakly inaccessible Suslin cardinal κλ must be a limit Suslin cardinal (and so λ = κ). Summarizing, our inaccessible Suslin cardinal κ is a limit of Suslin cardinals, ω and S(κ) has the scale property. In fact, S(κ) = ∃ω Γ, where Γ is a non-selfdual ω pointclass with o(Γ) = κ, Γ is closed under ∀ω , ∧, ∨, and Scale(Γ). It is possible that Γ = S(κ) if we are in the case of a Type IV hierarchy. We again note that these results can be obtained from just AD (cf. [Jac09]). We fix a Γ-complete set P (which exists by Wadge’s Lemma for all non-selfdual pointclasses under AD) and let {ϕn}n∈ω be a (regular) Γ-scale on P . An inspection of the standard argument shows that we have the following boundedness condition (as Γ is a boldface pointclass with the prewellordering property and closed under ω ∀ω and finite unions): any ∆ = Γ ∩ Γ˘ subset A of P is bounded in the codes, that is, sup{ϕx(x); n ∈ ω, x ∈ A} < κ. Our results from §§7 and 8 have to be understood in the context of proofs of partition properties for δ(∆) for highly closed pointclasses. For instance, consider the following example theorem as listed in [Jac08, Theorem 3.10]:

ω Theorem 14. Let Γ be non-selfdual, closed under ∀ω and finite unions, and with ω the prewellordering property. Define ∆ := Γ ∩ Γ˘. If ∃ω ∆ ⊆ ∆, then δ(∆) has the strong partition property. Note that if κ is an inaccessible Suslin cardinal and Γ is the pointclass defined as above, then Γ satisfies all of the requirements of Theorem 14, and therefore δ(∆) = κ has the strong partition property. Our results are extensions of this observation. The fact that κ has the strong partition property immediately implies that the ω ω-cofinal measure µ := Cκ on κ is a normal ultrafilter (cf. [Kle70, Theorem 2.1]). Finally, we recall one more result, due to Martin (cf. [Kec78, Theorem 3.7]) in the AD theory of pointclasses which will be used frequently later. For the sake of completeness, we sketch the proof.

ω Theorem 15 (Martin). Let Γ be a non-selfdual pointclass closed under ∀ω , ∧, ∨, and assume pwo(Γ). Let δ = δ(∆) (where ∆ = Γ ∩ Γ˘). Then ∆ is closed under unions and intersections of length < δ.

Proof. SAssume the contrary, and let ρ < δ be least such that some ρ union, say A = α<ρ Aα, of sets in ∆ that is not in ∆. Easily, ρ is regular. We may assume the Aα are strictly increasing. Since ρ < δ, there is a ∆-prewellordering of length ρ. The coding lemma then shows that A ∈ Γ˘ (since Γ˘ is closed under 8 ω ∃ω ). By assumption, A ∈ Γ˘ − ∆. Deﬁne a norm ϕ of length ρ on A by ρ(x) = least α such that x ∈ Aα. This is a Γ˘ norm on A, which shows that Γ˘ has the prewellordering property, a contradiction since for pointclasses Γ0 closed under ∧, ∨ we have the chain of implications pwo(Γ0) ⇒ Red(Γ0) ⇒ Sep(Γ˘ 0) and the separation property cannot hold on both sides of a non-selfdual pointclass. To see ∗ that ϕ0 is a Γ˘-norm, notice that the corresponding norm relation < can be written ∗ ∗ as x < y ↔ ∃α < ρ (x ∈ Aα ∧ y∈ / Aα). So, < is a ρ union of sets in ∆, and the coding lemma again shows that <∗ is in Γ˘. A similar computation works for ≤∗, showing ϕ is a Γ˘-prewellordering on A. q.e.d.

3. Reducing to base cases

Recall the 60 patterns mentioned in §1. A pattern [ x1 / x2 / x3 ] is called trivially inconsistent if there are 0 ≤ k < i < j ≤ 3 such that xi = ℵk and xj = ℵi. For example, [ ℵ0 / ℵ1 / M ] is trivially inconsistent. This is because ℵ1 is singular, but cf(ℵ2) = ℵ1, which is obviously impossible. A simple combinatorial calculation shows that there are 13 trivially inconsistent patterns. These are the patterns 13, 18, 33, 38, 44, 49, 51, 52, 53, 54, 55, 58, and 59 in our table of § 9. The remaining 47 patterns will be split into graphs according to the following rules: 0 • If P = [ x1 / x2 / x3 ] is a pattern with xi = M, and P = [ y1 / y2 / y3 ] is a pattern with yi = ℵ0 and for j 6= i, ½ xj if xj 6= ℵi, and yj = ℵ0 if xj = ℵi, then there is an edge from P to P 0. This corresponds to a forcing extension with Pˇr´ıkr´yforcing according to Theorem 1. • There is an edge from [ M / x2 / x3 ] to [ ℵ1 / x2 / x3 ]. This corresponds to a forcing extension adding ω1 many Cohen reals according to Theorem 2. Because of Theorems 1 and 2, if P is consistent and there is an edge from P to P 0, then P 0 is consistent. This allows us to reduce the consistency of patterns to the patterns that are top elements in the graph. We shall now list all components of this graph:

Base Case #1: [ M / M / M ].

[ M / M / M ] hhh q MM hhhh qqq MMM hhhh qq MM hhhh qq MM hs hhh qx qq MM& [ ℵ / M / M ] [ M / ℵ / M ] [ M / M / ℵ ] [ ℵ / M / M ] 0 M 0 M 0 1 M MM qq MMM qqq MM MMMqqq MMqq MMM qq MM qqqMMM MM qx qq MM& qx qq MM& MM& [ ℵ / ℵ / M ] [ ℵ / M / ℵ ] [ M / ℵ / ℵ ] [ ℵ / ℵ / M ] [ ℵ / M / ℵ ] 0 0 M 0 0 0 0 1 0 1 0 MM qqq qq MMM qq qqq MM qqq qq MM& qx qq qx qq [ ℵ0 / ℵ0 / ℵ0 ][ ℵ1 / ℵ0 / ℵ0 ] 9 The component of the graph reachable from the pattern [ M / M / M ] covers 12 of our patterns, the ones numbered 1, 5, 16, 20, 21, 25, 36, 40, 41, 45, 56, and 60 in our table.

Base Case #2: [ M / M / ℵ3 ].

[ M / M / ℵ3 ] q MM qqq MMM qqq MMM qx qq MM& [ ℵ0 / M / ℵ3 ] [ M / ℵ0 / ℵ3 ] [ ℵ1 / M / ℵ3 ] q MM qqq MMM qqq MMM qx qq MM& [ ℵ0 / ℵ0 / ℵ3 ][ ℵ1 / ℵ0 / ℵ3 ]

The component of the graph reachable from the pattern [ M / M / ℵ3 ] covers 6 of our patterns, the ones numbered 2, 17, 22, 37, 42, and 57. None of these was included in the component of Base Case #1.

Base Case #3: [ M / M / ℵ2 ].

[ M / M / ℵ2 ] q MM qqq MMM qqq MMM qx qq MM& [ ℵ0 / M / ℵ2 ] [ M / ℵ0 / ℵ0 ] [ ℵ1 / M / ℵ2 ] q MM qqq MMM qqq MMM qx qq MM& [ ℵ0 / ℵ0 / ℵ0 ][ ℵ1 / ℵ0 / ℵ0 ]

The component of the graph reachable from the pattern [ M / M / ℵ2 ] covers 6 of our patterns, the ones numbered 3, 20, 23, 40, 43, and 60. Of these, three were not included in the components of Base Cases #1 and #2.

Base Case #4: [ M / M / ℵ1 ].

[ M / M / ℵ1 ] q MM qqq MMM qqq MMM qx qq MM& [ ℵ0 / M / ℵ0 ] [ M / ℵ0 / ℵ1 ] [ ℵ1 / M / ℵ1 ] q MM qqq MMM qqq MMM qx qq MM& [ ℵ0 / ℵ0 / ℵ0 ][ ℵ1 / ℵ0 / ℵ1 ]

The component of the graph reachable from the pattern [ M / M / ℵ1 ] covers 6 of our patterns, the ones numbered 4, 19, 24, 39, 45, and 60. Of these, four were not included in the components of Base Cases #1 through #3. 10 Base Cases #5a-d: [ M / ℵ2 / x3 ].

[ M / ℵ2 / M ] q MM qqq MMM qqq MMM qx qq MM& [ M / ℵ2 / ℵ1 ] [ ℵ0 / ℵ2 / M ] [ M / ℵ2 / ℵ0 ] [ ℵ1 / ℵ2 / M ] MM q MM MMM qqq MMM MMM qqq MMM MM& qx qq MM& [ ℵ1 / ℵ2 / ℵ1 ][ ℵ0 / ℵ2 / ℵ0 ][ ℵ1 / ℵ2 / ℵ0 ]

[ M / ℵ2 / ℵ2 ] [ M / ℵ2 / ℵ3 ] MM q MMM qqq MMM qqq MM& qx qq [ ℵ0 / ℵ2 / ℵ2 ][ ℵ1 / ℵ2 / ℵ2 ][ ℵ0 / ℵ2 / ℵ3 ][ ℵ1 / ℵ2 / ℵ3 ]

This base case splits into four subcases, Base Case #5a [ M / ℵ2 / M ], Base Case #5b [ M / ℵ2 / ℵ3 ], Base Case #5c [ M / ℵ2 / ℵ2 ], and Base Case #5d [ M / ℵ2 / ℵ1 ]. The components of the graph reachable from the patterns [ M / ℵ2 / x3 ] cover 14 of our patterns, the ones numbered 6, 7, 8, 9, 10, 26, 27, 28, 29, 30, 46, 47, 48, and 50, none of which was included in the components of Base Cases #1 through #4.

Base Case #6: [ M / ℵ1 / M ].

[ M / ℵ1 / M ] q MM qqq MMM qqq MMM qx qq MM& [ ℵ0 / ℵ0 / M ] [ M / ℵ1 / ℵ0 ] [ ℵ1 / ℵ1 / M ] q MM qqq MMM qqq MMM qx qq MM& [ ℵ0 / ℵ0 / ℵ0 ][ ℵ1 / ℵ1 / ℵ0 ]

The component of the graph reachable from the pattern [ M / ℵ1 / M ] covers 6 of our patterns, the ones numbered 11, 15, 31, 35, 56, and 60. Of these, four were not included in the components of Base Cases #1 through #5.

Base Case #7: [ M / ℵ1 / ℵ3 ].

[ M / ℵ1 / ℵ3 ] q MM qqq MMM qqq MMM qx qq MM& [ ℵ0 / ℵ0 / ℵ3 ][ ℵ1 / ℵ1 / ℵ3 ]

The component of the graph reachable from the pattern [ M / ℵ1 / ℵ3 ] covers 3 of our patterns, the ones numbered 12, 32, and 57. Of these, two were not included in the components of Base Cases #1 through #6. 11 Base Case #8: [ M / ℵ1 / ℵ1 ].

[ M / ℵ1 / ℵ1 ] q MM qqq MMM qqq MMM qx qq MM& [ ℵ1 / ℵ1 / ℵ1 ][ ℵ0 / ℵ0 / ℵ0 ]

The component of the graph reachable from the pattern [ M / ℵ1 / ℵ1 ] covers 3 of our patterns, the ones numbered 14, 34, and 60. Of these, two were not included in the components of any of the other base cases.

By our earlier remarks, it is enough to show the consistency of the eight base cases in order to prove the consistency of all patterns that are not trivially inconsistent. Note that in some cases, the graph will not give us the optimal consistency strength upper bounds. For instance, the ZFC-pattern [ ℵ1 / ℵ2 / ℵ3 ] shows up in Base Case #5b and is obtained from the large cardinal pattern [ M / ℵ2 / ℵ3 ] by forcing. For more on upper and lower bounds, cf. § 9.

4. Base Cases #2, #5, and #7 In this section, we handle three of the base cases. These three are proved consistent with techniques from forcing with large cardinals and do not rely on either polarized partition properties or AD. Base Cases #5a-d are just Theorem 3 and do not need any large cardinals beyond the ones explicitly mentioned in the pattern that is created. The other cases in this section will be proved consistent from large cardinal assumptions by forcing in the following two theorems. None of these proofs is new. They all use published techniques and essentially consist of proof inspection to check that the relevant properties hold in the situation in which we are interested. Theorem 16 (Woodin). If there are κ < λ such that κ is supercompact and λ is measurable, then there is a model in which Base Case #2 holds (i.e., [ M / M / ℵ3 ]).

Proof. This theorem is discussed in [AH86, p. 591]. Theorem 1 of that paper is a generalization of Woodin’s result. Suppose V |= ZFC + “κ < λ are such that κ is supercompact and λ is measurable”. Let P0 be supercompact Radin forcing as defined in [AH86, p. 592sq], with κ playing the role of κ1 and λ playing the role of κ2. Let P1 = Col(ω, <κ), and let P = P0 × P1. Let G be P -generic over V, and take N as the choiceless inner model of [AH86, Theorem 1] defined with respect to G. By suitably modified versions of [AH86, Lemmas 1.1 through 1.4], N |= ZF+ “κ = ℵ1 is measurable via the club filter” + “λ = ℵ2 is measurable”. By the + + V appropriate version of [AH86, Lemma 1.2], N |= “λ = ℵ3 = (λ ) ”. Therefore, since V ⊆ N and V contains a λ+ sequence of subsets of λ, N does as well. This + means that N |= “λ = ℵ3 is not measurable”. q.e.d.

Theorem 17. If there is a supercompact cardinal, then there is a model of Base Case #7 (i.e., [ M / ℵ1 / ℵ3 ]). 12 Proof. This construction is essentially the same as in the proof of Theorem 16. Suppose V |= ZFC + GCH+ “κ < λ are such that κ is supercompact and +κ λ = κ ”. Again, let P0 be supercompact Radin forcing as in the proof of Theorem 16. Let P1 = Col(ω, <κ), and let P = P0 × P1. A similar argument as in the proof of Theorem 16, using GCH to show that λ can be symmetrically collapsed to become ℵ2, yields that the symmetric model N is such that N |= ZF + “κ = ℵ1 is measurable via the club ﬁlter” + “λ = ℵ2 is singular of coﬁnality κ = ℵ1” + + “λ = ℵ3 is not measurable”. q.e.d.

5. Base Cases #1, #3, #4, #6, and #8 In this section, we shall handle Base Cases #1, #3, #4, #6, and #8, all under the assumption that there is a model of AD. Among these, Base Case #3 is a special case since this is the famous AD-pattern:

Theorem 18 (Solovay-Martin). Assume AD. Then ℵ1 and ℵ2 are measurable cardinals and cf(ℵ3) = ℵ2.

Proof. Cf. [Kan94, Theorems 28.2, 28.6 and Corollary 28.8]. q.e.d.

For the next three of these base cases, we need a polarized partition property, relying heavily on the main theorem of § 8. Theorem 19. Assume AD. Then there is a limit cardinal κ such that the polarized partition property (κ, κ+, κ++) → (κ, κ+, κ++)κ holds.

Proof. By Proposition 12, there are weakly inaccessible Suslin cardinals. Now the result follows from our Theorem 41. q.e.d.

Theorem 20. If there is a model of AD, then there is a model of Base Case #4 (i.e., [ M / M / ℵ1 ]).

Proof. Using Theorem 19, we start with a limit cardinal κ such that (κ, κ+, κ++) → (κ, κ+, κ++)κ. By the proof of Lemma 6, for γ = κ, γ = κ+, or γ = κ++, γ → (γ)<ω1 . From this, it easily follows that γ → (γ)ω+ω, so by [Kle70, Theorem 2.1], + ++ κ, κ , and κ are all measurable. Forcing with Magidor-like forcing Pκ,κ++ , we obtain a model in which cf(κ++) = κ and the polarized partition relation (κ, κ+) → (κ, κ+)κ still holds (by the initial segment preservation from Lemma 9). Now we can collapse κ symmetrically to become ℵ1 and apply Theorem 4 to obtain our result. q.e.d.

Theorem 21 (Apter-Henle 1986). If there is a model of AD, then there is a model of Base Case #1 (i.e., [ M / M / M ]). 13 Proof. Cf. [AH86, Theorem 2]. The authors used (a slightly weaker version of) Kechris’ Theorem 24, listed as “personal communication” without a proof in [AH86]. q.e.d.

Theorem 22. If there is a model of AD, then there is a model of Base Case #6 (i.e., [ M / ℵ1 / M ]).

Proof. Again, from Theorem 19, we start with a limit cardinal κ such that (κ, κ+, κ++) → (κ, κ+, κ++)κ, and get with Lemma 6 the Ramsey-like property + ++ + ++ <ω1 (κ , κ ) → (κ , κ ) . Forcing with Magidor-like forcing Pκ,κ+ preserves the measurability of κ, as it does not add bounded subsets of κ+. Furthermore, by the weak ﬁnal segment preservation from Lemma 10, we preserve κ++ → (κ++)<ω1 , so κ++ stays measurable. By construction, we also have cf(κ+) = κ. Now, we can collapse κ symmetrically to become ℵ1 and apply Theorem 4 to obtain our result. q.e.d.

For Base Case #8, we rely on the methods of [AHJ00]. Theorem 23. If L(R) |= AD, then there is a model of Base Case #8 (i.e., [ M / ℵ1 / ℵ1 ]).

Proof. Suppose V is a model of V = L(R) and AD. We use the model N constructed and investigated in [AHJ00, §8] (in particular, [AHJ00, Theorem 8.1]) and applied in [AHJ00, Theorem 11.1]. In this model, which is a symmetric submodel of a forcing extension of V , ℵ2 and ℵ3 have coﬁnality ℵ1. Further, by [AHJ00, Proposition 6.2 and Lemma 8.2], N and V have the same bounded subsets of ℵ1. Thus, since V |=“ℵ1 is measurable”, N |=“ℵ1 is measurable” as well. This means that N is as desired. q.e.d.

6. Kechris’ Theorem In this section, we shall prove Kechris’ theorem, announced in the 1980s, but not published. The proofs of our extensions of this theorem in §§7 and 8 build on this proof and will use definitions from this section. Theorem 24. Assume AD and let κ be a weakly inaccessible Suslin cardinal. Then + ++ + ++ ϑ for all ϑ < ω1 we have (κ, κ , κ ) → (κ, κ , κ ) . Throughout this section, κ will be a weakly inaccessible Suslin cardinal (which exists by Proposition 12). Note that by Fact 8 it doesn’t matter whether we are using the standard or the c.u.b. version of the partition property, and we shall freely switch between them. Partition property proofs under AD always follow the same lines as abstracted by Tony Martin (cf. [Kec78, Lemma 11.1] and [Jac09, Theorem 2.3.4]): to show κ → κλ we must find a sufficiently good coding of the functions f : λ → κ. This involves identifying a Lévypointclass Γ and a coding map ϕ: ωω → P(λ × κ) with certain coding relations being in ∆. In this paper we shall use Martin’s method directly, so the reader need not be familiar with these general results. 14 In our setting, we have already identified the pointclass Γ in our discussion in §2.4: it is the (Steel) pointclass forming the lowest level of the projective-like hierarchy containing S(κ). We have seen that this pointclass has the required ω properties: S(κ) = ∃ω Γ has the scale property. The pointclass Γ (possibly Γ = ω S(κ)) is scaled, non-selfdual, closed under ∀ω and finite intersections and unions. We fixed a Γ-complete set P and a regular Γ-scale {ϕn}n∈ω on P which allows boundedness arguments. In the following, P and ϕ will be used to code ordinals less than κ. Since we also want to code higher ordinals, we shall have to come up with a means of coding for these (in §6.2). By µ, we denote the the ω-cofinal measure on κ (which is an ultrafilter by + + ++ Theorem 14). We shall show that [α 7→ α ]µ = κ and that δ := [α 7→ α ]µ = κ++. The first claim can be proved directly (Lemma 29), after which we shall show the following auxiliary theorem:

+ + ϑ Theorem 25. (κ, κ , δ) → (κ, κ , δ) for all ϑ < ω1. ϑ It follows immediately from Theorem 25 that δ → (δ) for all ϑ < ω1. In particular, δ is regular. By showing that κ+ < δ ≤ κ++ (Claim 34), we establish that δ = κ++, thus proving Theorem 24. 6.1. Countable unions of <α-Suslin sets. An ordinal α < κ is called ~ϕ-strongly reliable if for all β < α, we have sup{ϕn(x); n ∈ ω ∧ ϕ0(x) ≤ β} < α. Let C ⊆ κ be a c.u.b. set contained in the ~ϕ-strongly reliable ordinals. Without loss of generality, we may assume that C is contained in the Suslin cardinals. The relation R(x, y) ↔ x, y ∈ P ∧ ϕ0(x) ≤ ϕ0(y) is in Γ, and so admits a Γ-scale ~σ (with norms into κ). By boundedness, we may assume C has the property that for all α ∈ C and β < α, if R(x, y) and ϕ0(x) ≤ ϕ0(y) ≤ β, then supn σn(x, y) < α. Let Cω denote the elements of C of cofinality ω. α As in §2.4, for α ∈ Cω, let Σ0 denote the pointclass of countable unions of S α α α α sets which are in {S(β); β < α}. Thus, Scale(Σ0 ). Define Σn, Πn from Σ0 as α α usual. Then Scale(Π1 ), and Σ1 is the pointclass of α-Suslin sets. From the Coding Lemma and the Kunen-Martin Theorem 11 it follows that α+ is the supremum of α the lengths of the Σ1 prewellorderings, and is the supremum of the lengths of the α + Σ1 wellfounded relations. In particular, α is regular. α 1 The pointclass Σ1 behaves sufficiently similar to Σ1 to allow proving the following result (by essentially the same argument as is used in the countable partition property of ℵ1, cf. [Kec78, Theorem 11.2]): + + ϑ Theorem 26. For all ϑ < ω1, we have α → (α ) . Applying [Kle70, Theorem 2.1] again, we get that the ω-cofinal normal measure ω + µα := Cα+ on α is an ultrafilter. We shall use this measure in Lemma 33 and its proof. As we shall need to do arguments about α uniformly, we check in the next result that the assignment of scales and universal sets is uniform:

α α Lemma 27. There is a function α 7→ (A , ~ρ ) which assigns to each α ∈ Cω a α α α α α α universal Σ0 set A and a Σ0 scale ~ρ = {ρn}n∈ω on A with norms into α. Furthermore, there is a function α 7→ (Bα, V α) which assigns to such α a universal α α α α α Σ1 set B and a tree V on ω × α with B = p[V ]. Finally, there is a function α ~α α α ~ α α 7→ (Q , ψ ) which assigns to such α a universal Π1 set and a Π1 -scale ψ on Q . 15 α Proof. For α ∈ Cω let R = {hx, yi ; x, y ∈ P ∧ ϕ0(x) ≤ ϕ0(y) < α}. From the definition of C, Rα can be written as an α union of sets each of which is <α- α α α Suslin. Thus, R ∈ Σ0 . Since R is a prewellordering of length α, it cannot be α α <α Suslin. Define A = {hτ, zi ; ∃n((τ(z))n ∈ R )}, where we view every real τ as a strategy for II in an integer game in some standard manner, and τ(z) is the α α α result of applying τ to z. Clearly A ∈ Σ0 . Moreover, since R has Wadge degree α α at least α, it follows easily that A is Σ0 -universal. We define, uniformly in α, a α α α α tree U with A = p[U ]. Define (s, (α0, . . . , αn)) ∈ U iff

(i) α0 > max{α1, . . . , αm}, and α0 < α. (ii) α1 ∈ ω ω (iii) There is a hτ, zi ∈ ω extending s such that if (τ(z))α1 = hx, yi, then σi(x, y) = αi+2 for all i ≤ n − 2 (~σ is the scale on R as above). α α α From the closure properties of C it follows that A = p[U ]. Let then {ρ¯n} α α be the semi-scale derived from the Suslin representation U , and let {ρn} be α α the corresponding scale. Since eachρ ¯n maps into α, so does ρn, using property (i) in the deﬁnition of Uα [In passing from the semi-scale to the scale we can take α α α ρn(w) = |(¯ρ0 (w),..., ρ¯n(w))|≺, where ≺ is lexicographic ordering on the set on α α n + 1 tuples satisfying (i).] To see that ~ρ is a Σ0 -scale, it is enough to show that α α α the semi-scale {ρ¯n} is a Σ0 semi-scale, since Σ0 is closed under ∧, ∨. However, ∗ ∗ α each of the norm relations

6.2. Coding of ordinals below κ, κ+ and δ. The coding of elements of κ is completely standard: A real x will code an ordinal below κ iff x ∈ P . In this case, x codes the ordinal |x| = ϕ0(x). We let P0 = P be the set of codes of ordinals below κ. In order to code ordinals less than κ+, we need a tree T + on ω × κ which we shall use in our coding of the ordinals. For the definition of T +, we need a number of auxiliary objects: W , T2, and U. ω Let W = {w ∈ ω ; ∀n (w)n ∈ P }. Define the norm ψ on W by ψ(w) = sup{ϕ0((w)n); n ∈ ω}. It is easy to see that ψ is a Γ-norm on the set W ∈ Γ. If we define a tree T2 on ω × κ by (s, (α0, . . . , αn)) ∈ T2 iff there is a w extending s such that

w ∈ W ∧ ∀i ≤ n (ϕi0 ((w)i1 ) = αi), then p[T2] = W . For α ∈ C with cf(α) = ω we also have that p[T2¹α] = Wα := {w ∈ W ; ψ(w) ≤ α}. Furthermore, we define a tree U on (ω)4 ×κ×κ as follows (we recycle the notation here; U has nothing to do with the trees U α above). As a motivation, it is helpful 16 to think of the first two coordinates of U in the definition that follows as defining ψ(x) ψ(x) reals x, y with x ∈ W and y defining a Σ1 relation via the universal set B from Lemma 27. Define (s, t, u, v, ~α, β~) ∈ U iff there are x, y, z, w ∈ ωω extending s, t, u, v such that:

(i) z codes the reals z0, z1,... , w codes w0, w1,... , and for each i, n ∈ ω the subsequence γj = αhi,n,ji of the ~α satisfies the following. View y as coding a Lipschitz integer strategy for II, and set ri = hy(hzi, zi+1i), wii. Let b = (ri)n. Then (b,~γ) ∈ [T~σ], where T~σ is the tree of the scale ~σ on R. (ii) For each i, n ∈ ω we have (using the notation immediately above) if δj = ~0 ~0 βhi,n,ji then δ0 ∈ ω and (h(b)1, xδ0 i, δ ) ∈ [T~σ], where δ = (δ1, δ2,... ). Let us explain the idea behind the definition of U: the objects w, ~α, β~ are ψ(x) attempting to witness that the z0, z1,... form a decreasing sequence in the Σ1 relation coded by y, as in the proof of the Kunen-Martin Theorem 11. The relation ψ(x) coded by y is the set of (c, d) such that ∃w ∀n hy(hc, di), win = (e, f) ∈ R (where R is as in the proof of Lemma 27). The β~ ordinals witness that the various ψ(x) f reals satisfy ϕ0(f) < ϕ0((x)n) for some n, and so (e, f) ∈ R . Lemma 28. Suppose that x ∈ W , ψ(x) ∈ C, and y codes a wellfounded relation ψ(x) A in Σ1 . Then Ux,y is wellfounded. Furthermore, |Ux,y¹ψ(x)| ≥ |A|. ~ ψ(x) Proof. If (z, w, ~α, β) ∈ [Ux,y], then for each i, A(zi, zi+1), where A is the Σ1 ψ(x) ~ relation coded by y: A(c, d) ↔ ∃w ∀n hy(hc, di), win = (e, f) ∈ R The β witness that the various (e, f) are in Rψ(x). So, as the Kunen-Martin Theorem 11, this produces an infinite decreasing chain through A, a contradiction. For any c, d such that A(c, d), we can find a w such that for all n, if hy(hc, di), win = (e, f), then supj σj(e, f) < ψ(x) and supj σj(f, xk) < ψ(x) for any k such that ϕ0(xk) > ϕ0(f). This follows from the closure properties of C and the fact that ψ(x) ∈ C. The proof of the Kunen-Martin Theorem 11 then shows that Ux,y¹ψ(x) has rank at least |A|. q.e.d.

We note that it is important for the following argument that for x, y as in the statement of Lemma 28 that the entire tree Ux,y is wellfounded (not just Ux,y¹ψ(x)). Finally, we can now define the tree T + on (ω)2 × κ × (ω)4 × κ × κ such that ~ + ~ (a, b,~γ, s, t, u, v, ~α, β) ∈ T if and only if (b,~γ) ∈ T2,(s, t, u, v, ~α, β) ∈ U, and there are σ, r, x, y extending a, b, s, t, respectively, such that σ(r) = hx, yi. We may identify T + with a tree on ω × κ by identifying the last coordinates with a single coordinate (i.e., taking a reasonable bijection between ω × κ × (ω)4 × κ × κ and κ). We furthermore fix a reasonable bijection between κ and κ<ω. We assume (without loss of generality) that our c.u.b set C is closed under both of these bijections. Coding ordinals below κ+. A code for an ordinal below κ+ will be a pair (x, σ) ω where x ∈ P and thus codes an ordinal |x| = ϕ0(x) below κ, and σ ∈ ω such that + <ω Tσ is wellfounded. Using our bijection between κ and κ , we can ask for the rank + + of an ordinal ξ < κ in the tree Tσ ; we write |Tσ (ξ)| for this. Given a pair (x, σ), we + now consider the map f : κ → κ defined by α 7→ |(Tσ ¹α)(|x|)|. Using the ω-cofinal ω normal measure µ := Cκ on κ, we now define |(x, σ)| := [f]µ. We let P1 be the set of codes of ordinals below κ+. The next lemma shows that this works. + + + Lemma 29. {|(x, σ)| ;(x, σ) ∈ P1} = κ . Also, κ = [α 7→ α ]µ. 17 + Proof. Suppose (x, σ) ∈ P1, and let f = fx,σ be given by f(α) = |Tσ ¹α(|x|)| for ∗ ∗ all α > |x|. If g : κ → κ is such that ∀µα g(α) < f(α), then ∀µα ∃β < α g(α) = 0 |tσ¹α(β)|. By normality of µ we may fix β < κ, and fix x ∈ P0 coding β, such ∗ + 0 0 that ∀µα g(α) = |(Tσ ¹α)(|x |)|. So, [g]µ = |(x , σ)|. So, the ordinals coded by P1 form an initial segment of the ordinals. This argument also shows that there + is a map from κ = [α 7→ α]µ onto |(x, σ)|, namely β 7→ [α 7→ |Tσ ¹α(β)|]µ. So, + {|(x, σ)| ;(x, σ) ∈ P1} ≤ κ . + 0 If ≺ is a wellorder of κ, let f≺ be given by f≺(α) = |≺¹α| < α . If |≺| = |≺ |, 0 then there is a c.u.b. subset of κ on which f≺ and f≺0 agree. Also, if |≺| < |≺ |, then there is a c.u.b. set on which f≺(α) < f≺0 (α). This gives an order-preserving + + + + map from κ into [α 7→ α ]µ. So, [α 7→ α ]µ ≥ κ . + Suppose γ = [f]µ where f(α) < α for all α < κ. Consider the following game Gf Player I r(0) r(1) r(2) ... Player II x(0) x(1) x(2) ... y(0) y(1) y(2) ... where player I plays out a real r ∈ ωω and player II plays out reals x, y. We interpret the real r as coding countably many reals {(r)i ; i ∈ ω} and x as coding {(x), ; i ∈ ω}. Let i be least, if it exists, such that (r)i ∈/ P0 or (x)i ∈/ P0. Player I loses if ri ∈/ P0. Assume then that (r)i, (x)i ∈ P0 for all i. Thus, x ∈ W . Recall ψ(x) ψ(x) = supi |(x)i|. Player II then wins iff ψ(x) ≥ ψ(r) and y codes a Σ1 wellfounded relation Ay of rank > f(ψ(x)). Player I cannot have a winning strategy σ, for suppose σ were winning for ω ω player I. Note that {(r)0 ; r ∈ σ[ω × ω ]} ⊆ P0. By boundedness, let α0 ∈ C be ω ω such that α0 > sup{|(r)0| ; r ∈ σ[ω × ω ]}. Fix a real x0 ∈ P0 with |x0| = α0. Note then that {r1 ; r ∈ σ[{(x, y);(x)0 = x0} ⊆ P0. By boundedness, fix α1 > α0 in C with α1 > sup{|(r)1| ; r ∈ σ[{(x, y);(x)0 = x0}. Continuing, we define αi, ω and xi for all i. Let x ∈ ω be the real with (x)i = xi for all i. Let α = supi αi. α Clearly, α ∈ C. By the coding lemma, there is a Σ1 wellfounded relation, say coded by the real y, of length greater than f(α). If player II plays x and y, then player II defeats σ, a contradiction. + Let now σ be a winning strategy for player II in Gf . First note that Tσ is ~ + wellfounded. For a branch (r,~γ, x, y, z, w, ~α, β) ∈ [Tσ ] would give r ∈ W (witnessed by ~γ) and so x ∈ W and ψ(x) ≥ ψ(r) as σ(r) = hx, yi and σ is winning for player II. ψ(x) Also, y codes a Σ1 wellfounded relation Ay of rank at least f(ψ(x)). z would however give a decreasing chain in Ay, a contradiction. Next note that there is a c.u.b. D ⊆ κ such for all α ∈ D and r ∈ W with ψ(r) = α, if σ(r) = hx, yi, then ψ(x) = ψ(r). This follows by a boundedness argument similar to the above. We may assume D ⊆ C. For α ∈ D with cf(α) = ω, let r ∈ W with ψ(r) = α. If α σ(r) = hx, yi, then w ∈ W and ψ(x) = ψ(r) = α. Thus, y codes a Σ1 wellfounded + relation Ay of rank at least f(α). From Lemma 28 it follows that |Tσ ¹α| > f(α). + So, [f]µ ≤ [α 7→ |Tσ ¹α|]µ. It follows that for some x ∈ P that [f]µ = |(x, σ)|. So, + + [α 7→ α ]µ ≤ {|(x, σ)| ;(x, σ) ∈ P1} ≤ κ . q.e.d.

Coding ordinals below δ. We use a variation T ++ of the tree T + above. The tree T ++ will be defined exactly as T + except we use a tree U 0 in place of U. In 18 0 order to define U , we need an auxiliary tree V2. In Lemma 31, we shall construct a tree V which is then refined to V2 in Lemma 32. α ~α First, recall from Lemma 27 that there is a function α 7→ (Q , ψ ), for α ∈ Cω, α α ~α α α where Q is a universal Π1 set, and ψ is a (regular) Π1 -scale on Q . Note that α + + α the norms ψn map into α , since α is the supremum of the lengths of the Σ1 wellfounded relations. We need the following technical lemma. Lemma 30. There is a continuous function c: ωω × ωω → ωω such that for all x ∈ W , β ≥ ψ(x), and y ∈ ωω we have that c(x, y) ∈ Qβ iff there is no z such that ψ(x) ψ(x) β for all n, we have B (hy, hzn, zn+1ii). Here, B and Q are as in Lemma 27.

Proof. We have (the ﬁrst line below deﬁnes E):

ψ(x) E(x, y) ↔ ¬∃z ∀i B (hy, hzi, zi+1ii) ψ(x) ↔ ¬∃z ∀i ∃w ∀n R (y(hzi, zi+1i, w, n)) ψ(x) ↔ ¬∃u ∀m R (y(h(u0)m0 , (u0)m0+1i, (u1)m0 , m1))

↔ ¬∃hu, vi ∀m (R(a, b) ∧ R(b, xv(m))), where

a = a(y, u, m) = (y(h(u0)m0 , (u0)m0+1i, (u1)m0 , m1))0,

b = b(y, u, m) = (y(h(u0)m0 , (u0)m0+1i, (u1)m0 , m1))1. ψ(x) We use here the fact that (for all j) R(a, b) ∧ R(b, xj) holds iﬀ R (a, b) ∧ ψ(x) β β R (b, xj) iﬀ R (a, b) ∧ R (b, xj). We therefore have

β β E(x, y) ↔ ¬∃t = hu, vi ∀m (R (a, b) ∧ R (b, xv(m))) β ↔ ¬∃t ∀m R (c0(x, y)(c1(x, y), t, m)) ↔ Qβ(c(x, y)) where c(x, y) = hc0(x, y), c1(x, y)i and c0, c1 are continuous functions such that m m c0(x, y) is a strategy for II satisfying c0(x, y)(c1(x, y), t, m) = ha(y, u, 2 ), b(y, u, 2 )i m if m is even and c0(x, y)(c1(x, y), t, m) = hb(y, u, 2 ), xv(m)i if m is odd. We may take c1(x, y) = hx, yi, and then easily get a continuous c0 satisfying this equation. q.e.d.

Lemma 31. There is a tree V on ω × ω × κ × κ such that (x, y) ∈ p[V ] iﬀ x ∈ W ψ(x) and y codes a Σ1 wellfounded relation. Furthermore, if x ∈ W , ψ(x) ∈ C, and ψ(x) + y codes a wellfounded Σ1 relation, then there is a β < ψ(x) such that Vx,y¹β α is illfounded. In fact, for any α ∈ C with cf(α) = ω, and any A ∈ Σ1 consisting of ψ(x) pairs (x, y) such that x ∈ W , ψ(x) ≤ α, and y codes a wellfounded Σ1 relation, there is a β < α+ such that A ⊆ p[V ¹β].

Proof. Deﬁne (s, t, ~α, β~) ∈ V iﬀ ~ (i) β0 ∈ C and β0 > max{~α, β}. (ii) (s, ~α) ∈ T2. 19 β0 (iii) There are x, y extending s, t such that βi = ψi (c(x, y)) for 1 ≤ i < lh)s), β0 β0 where {ψi } is the scale on Q from Lemma 27 and c is the continuous function of Lemma 30. It is clear that V has the desired properties from Lemma 30. For the last property α α α α claimed, we use the fact that {ψi } is a Π1 -scale, and so every Σ1 subset of Q is bounded in these norms. q.e.d.

The next lemma is a small variation of the previous one, allowing y to code countably many wellfounded relations instead of just one.

Lemma 32. There is a tree V2 on ω × ω × κ × κ such that (x, y) ∈ p[V ] iﬀ x ∈ W ψ(x) and for all n,(y)n codes a Σ1 wellfounded relation. Furthermore, if x ∈ W , ψ(x) ∈ C then there is a c.u.b. D ⊆ α+ such that for all γ ∈ D and all y such that ψ(x) for all n,(y)n codes a wellfounded Σ1 relation of length less than γ, Vx,y¹γ is illfounded.

Proof. The tree V2 is constructed as V except that in (iii) we require that β = ψβ0 (c(x, (y) )). Given α ∈ C with cf(α) = ω, deﬁne D ⊆ α+ as fol- i i0 i1 + α lows. For β < α , let Bα,β = {(x, y); x ∈ W ∧ ψ(x) ≤ α ∧ y codes a Σ1 α + wellfounded relation of length ≤ β}. Since ∆1 is closed under <α unions and α intersections (by the usual Martin argument), it follows that Bα,β ∈ ∆1 . Let α + g(α, β) = sup{ψn (c(x, y)) ; (x, y) ∈ Bα,β} < α by boundedness. Let D be the c.u.b. sets of points below α+ which are closed under g. q.e.d.

We define (s, t, u, ~α, β,~ v, a, b,~γ, ~δ) ∈ U 0 iff there are x, τ, w, y, z, w extending s, t, u, v, a, b, respectively, such that ~ (i) (s, u, ~α, β) ∈ V2 (ii) τ(w) = y (iii) (s, v, a, b,~γ, ~δ) ∈ U. Here U is as in Lema 28. We now define the tree T ++ on ω2 × κ × ω3 × κ2 × ω3 × κ2 consisting of tuples (c, d, ~η, s, t, u, ~α, β,~ v, a, b,~γ, ~δ) such that there are σ, r, x, τ extending c, d, s, t such that: (i) σ(r) = hx, τi (ii) (d, ~η) ∈ T2. (iii) (s, t, u, ~α, β,~ v, a, b,~γ, ~δ) ∈ U 0. Let us explain the definition of T ++: The first coordinate of the tree T ++ produces a strategy σ. We intend for σ to be a strategy such that when player I plays r ∈ W , then σ(r) = hx, τi where x ∈ W and ψ(x) ≥ ψ(r). The object τ is also a strategy which we intend to do the following. If player I plays a w such ψ(x) that for all n,(w)n codes a Σ1 wellfounded relation (so w codes the ordinal supn{|(w)n|}, where |(w)n| means the rank of the relation coded by (w)n), then ψ(x) ++ τ(w) = y codes a wellfounded relation in Σ1 . Finally, T attempts to produce an infinite decreasing chain in the relation coded by y, as in the Kunen-Martin Theorem 11. + For the following result, remember that µα is the ω-cofinal measure on α which exists by Theorem 26. 20 + ++ Lemma 33. For all α ∈ Cω we have jµα (α ) = α . Proof. The proof follows by a Kunen tree argument, as in the proof for the odd projective ordinals. It is also a special case of the argument given below. Briefly, define the tree K on ω2 × α+ × ω3 × α2 by: (s, t, ~α, u, v, w, β,~γ~ ) ∈ K iff there are τ, w, and y extending s, t, u such that τ(w) = y,(t, ~α) ∈ V¯ , and (t, u, v, w, β,~γ~ ) ∈ U. ¯ ¯ α Here V is a tree such that p[V ] is the set of w such that for all n,(w)n codes a Σ1 wellfounded relation, and there is a c.u.b. D ⊆ α+ such that for all β ∈ D there ¯ is a w ∈ p[V ¹β] such that for all n,(w)n has rank |(w)n| < β and supn |(w)n| = β (we can take V¯ to be a section of the tree V2 from Lemma 32). If F : α+ → α+, consider the game where player I plays out w, player II plays α out y, and player II wins iff whenever for all n,(w)n codes a Σ1 wellfounded α relation, then y codes a Σ1 wellfounded relation of length > f(|w|, where |w| is the supremum of the lengths of the relations coded by the (w)n. By boundedness, player II has a winning strategy σ for the game. For any β ∈ D with cf(β) = ω we ¯ + + ++ have f(β) < |Vσ¹β|. This shows [f]µα < α , and so jµα (α ) ≤ α . The lower bound follows from the embedding argument given earlier (the second paragraph of the proof of Lemma 29). q.e.d.

++ ++ Claim 34. [α 7→ α ]µ ≤ κ .

++ Proof. Fix f : κ → κ such that for all α, f(α) < α . Consider the game Gf deﬁned as follows: Player I r(0) r(1) r(2) ... Player II x(0) x(1) x(2) ... τ(0) τ(1) τ(2) ...

If there is a least i such that (r)i or (x)i is not in P0, then player I wins iﬀ (r)i ∈ P0. Suppose then that r, x ∈ W , that is, for all n,(r)n ∈ P0 ∧ (x)n ∈ P0. Let α = ψ(x) = supn ϕ0((x)n). Then player II wins provided τ is a strategy with the + + α following properties. There is a g : α → α such that if for all n,(w)n codes a Σ1 α wellfounded relation, then σ(w) codes a Σ1 wellfounded relation of length > g(|w|), and also [g]µα ≥ f(α). Here |w| is the supremum of the lengths of the wellfounded relations coded by the (w)n. The usual boundedness argument and Lemma 33 (and its proof) show that player I cannot have a winning strategy for Gf . Let σ be a winning strategy for ++ ++ player II in Gf . Inspecting the deﬁnition of T shows that Tσ is wellfounded. There is a c.u.b. C2 ⊆ C such that if player I plays r ∈ W with ψ(r) = α ∈ C2, then σ(r) = hx, τi where x ∈ W and ψ(x) = α. Let α ∈ C2 with cf(α) = ω. Fix r ∈ W with ψ(x) = α, and let σ(r) = hx, τi. Then there is a c.u.b. D ⊆ α+ such that for β ∈ D with cf(β) = ω, there is a w such that for all n,(w)n codes α a Σ1 wellfounded relation, supn |(w)n| = β, and (V2)x,w¹β is illfounded. Also, for α such β and w, τ(w) = y codes a Σ1 wellfounded relation of length > g(β), where ++ [g]µα > f(α). It follows that for such α that [β 7→ Tσ ¹β]µα > f(α). 0 From the normality of the measures µα it follows that if [f ]µ < [f]µ, then there + 0 ++ is a function h such that h(α) < α and f (α) = [β 7→ Tσ ¹β(h(α))| for almost all + + ++ α. This shows that [f]µ is a wellordering of [α 7→ α ]µ = κ . So, [f]µ < κ . So, ++ ++ [α 7→ α ]µ ≤ κ . q.e.d.

21 ++ ++ Let δ = [α 7→ α ]µ. We have shown δ ≤ κ . The lower bound will follow from the fact that δ is regular, which follows from the partition property δ → (δ)ϑ for ϑ < ω1 which follows from the polarized partition property we show below. We are ﬁnally in the position to code ordinals below δ. Such a code is a triple + ++ of the form (x, σ1, σ2), where x ∈ P0, Tσ1 is wellfounded, and Tσ2 is wellfounded. Let P2 be the set of codes for ordinals below δ. + Since (x, σ1) ∈ P1, it determines a function h with h(α) < α almost everywhere. ++ The triple then codes the ordinal [f]µ where f(α) = [β 7→ |Tσ2 ¹β(h(α)|]µα .

Lemma 35. Every ordinal below δ is coded by a triple in P2.

Proof. Clear from the proof of Claim 34. q.e.d.

6.3. Proof of Theorem 25. Let P be a partition of the block functions from + 3 × ω · ϑ to (κ, κ , δ). Fix a bijection π : ω · ϑ → ω. Let ≺π be the corresponding wellordering of ω.An ordinal j < ω · ϑ can be identified with a pair j = (i, n) where i < ϑ and n < ω, using lexicographic ordering on the pairs. We shall frequently pass back and forth from this identification. Consider the following game G, where player I plays out a real hx, y, zi, and 0 0 0 player II plays out the real hx , y , z i. If there is an j < ω · ϑ such that (x)π(j) ∈/ P0 0 or (x )π(j) ∈/ P0, then player I wins iff for the least such j we have that (x)π(j) ∈ P0. 0 Suppose then that for all j < ω · ϑ,(x)π(j) and (x )π(j) are in P0. In this case, x and x0 each determine a function from ω · ϑ to κ. Namely, x codes the function ¯ 0 ¯ Fx(j) = |(x)π(j)|. Likewise, x codes the function Fx0 . So, together they produce a ¯ ¯ function F = Fx,x0 : ϑ → κ given by F (i) = supn max{Fx(i, n), Fx0 (i, n)}. Suppose next that there is an α < κ such that one of the following holds. + + (a) There is a j < ω · ϑ such that either T ¹α or T 0 ¹α is illfounded. (y)π(j) (y )π(j) + ++ ++ (b) There is a β < α and a j < ω · ϑ such that either T ¹β or T 0 ¹β (z)π(j) (z )π(j) is illfounded. Let α < κ be least such that (a) or (b) above holds. If (a) holds, let j be least such that (a) holds for α and this j. In this case, Player I wins provided T + (y)π(j) is wellfounded. If (a) does not hold at α, but (b) does, let (β, j) be lexicographically least such that (b) holds. Player I wins in this case provided T ++ ¹β is (z)π(j) wellfounded. Suppose finally that neither (a) nor (b) hold for all α < κ. Then each of y, y0 determine a block function from (ω·ϑ)×κ to κ. Namely, for α ∈ C and j < ω·ϑ, let + 0 g¯y(= |T ¹α|. Likewise, y determines the block functiong ¯y0 . Together, they de- (y)π(j) termine the block function g : ϑ × κ → κ by g(α, i) = supn max{g¯y(α, j), g¯y0 (α, j)}, + where j = (i, n). Finally, g determines a function G = Gy,y0 : ϑ → κ by G(i) = [α 7→ g(α, i)]µ. 0 ¯ ¯ In a similar fashion, each of z, z determine block functions hz, hz0 . For α ∈ C, + ¯ ++ ¯ β < α , and j < ω · ϑ, let hz(α, β, j) = |T ¹β|. Similarly define hz0 . Together (z)π(j) they determine a block function h defined by: for α ∈ C, β < α+, and i < ϑ, ¯ ¯ let h(α, β, i) = supn max{hz(α, β, j), hz0 (α, β, j)}, where j = (i, n). Finally, h 0 determines a function H = Hz,z : ϑ → δ given by H(i) = [α 7→ [β 7→ h(α, β, i)]µα ]µ. Finally in this case we say Player II wins the run of the game iff P(F, G, H) = 1. 22 Suppose without loss of generality that Player II has a winning strategy τ (the case where player I has a winning strategy is slightly easier). We define first a c.u.b. set C0 ⊆ κ. For each η < κ and j < ω · ϑ, let 0 Aη,j = {(x, y, z); ∀j ≤ j ((x)π(j0) ∈ P0 ∧ ϕ0((x)π(j0)) ≤ η}.

Clearly Aη,j ∈ ∆ (recall Γ is the Steel pointclass of Wadge rank κ). Since τ is 0 0 0 0 winning for Player II, if (x, y, z) ∈ Aη,j, and τ(x, y, z) = (x , y , z ), then ∀j ≤ 0 j ((x )π(j0) ∈ P0) and by boundedness 0 0 0 0 0 ρ0(η, j) := sup{ϕ0((x )π(j0));(x , y , z ) ∈ τ[Aη,j] ∧ j ≤ j} < κ.

Let C0 ⊆ C be c.u.b. and closed under ρ0. + + We next deﬁne a c.u.b. C1 ⊆ κ . For α ∈ C0 with cf(α) = ω, η < α , and j < ω · ϑ, let

Aα,η,j ={(x, y, z); ∀j ((x)π(j) ∈ P0 ∧ ϕ0((x)π(j)) < α) ∧ ∀α0 < α ∀β < (α0)+ ∀j (T + ¹α and T ++ ¹β are wellfounded) (y)π(j) (z)π(j) ∧ ∀j0 ≤ j (|T + ¹α| ≤ η)}. (y)π(j0) 0 0 0 Since τ is winning for player II, if (x, y, z) ∈ Aα,η,j and τ(x, y, z) = (x , y , z ) then 0 0 α x ∈ W and ϕ0((x )π(j)) < α for all j. Furthermore, since Aα,η,j ∈ ∆1 , we have by boundedness that + 0 0 0 0 + ρ1(α, η, j) := sup{|T 0 ¹α| ; j ≤ j ∧ (x , y , z ) ∈ τ[Aα,η,j ]} < α . (y )π(j0) Construct (uniformly in α) sets Dα ⊆ α+ which are c.u.b. and closed under (η, j) 7→ + ∗ α + ρ1(α, η, j). Let E1 ⊆ κ be the set of [f]µ such that ∀µα f(α) ∈ D . Let F1 ⊆ κ + + be the set of limit points of ordinals of the form [α 7→ |Tx ¹α|]µ where Tx is + + wellfounded. F1 is c.u.b. in κ from Lemma 29. Clearly E1 is also c.u.b. in κ . Let C1 = E1 ∩ F1. + Finally, we deﬁne a c.u.b C2 ⊆ δ. For α ∈ C0 with cf(α) = ω, β, η < α , and j < ω · ϑ, let

Aα,β,η,j ={(x, y, z); ∀j ((x)π(j) ∈ P0 ∧ ϕ0((x)π(j)) < α) ∧ ∀α0 < α ∀β < (α0)+ ∀j (T + ¹α and T ++ ¹β are wellfounded) (y)π(j) (z)π(j) + 0 0 ++ ∧ ∀j |T ¹α| < β ∧ ∀(β , j ) ≤lex (β, j)(|T ¹β| ≤ η)}. (y)π(j) (z)π(j) α We have Aα,β,η,j ∈ ∆1 . Since τ is winning for Player II, for each (x, y, z) ∈ Aα,β,η,j , 0 0 0 0 0 ++ if τ(x, y, x) = (x , y , z ) then ∀(β , j ) ≤lex (β, j) T 0 ¹β is wellfounded. By (z )π(j0) boundedness, ++ 0 0 0 0 + ρ2(α, β, η, j) := sup{|T 0 ¹β| ;(x , y , z ) ∈ τ[Aα,β,η,j ] ∧ j ≤ j} < α . (z )π(j0) α + α ++ Let E be a c.u.b. subset of α closed under ρ2. Let E2 ⊆ α be the c.u.b. set α of all [f]µα where ran(f) ⊆ E . Let E2 ⊆ δ be the c.u.b. set of all [g]µ where α g(α) ∈ E2 for α < κ. E2 is c.u.b. in δ from from the deﬁnition of δ. Let F2 be the ++ c.u.b. subset of δ consisting of limits of points of the form [α 7→ [β 7→ |Tx ¹β|]µα ]µ ++ where Tx is wellfounded. From Claim 34, F2 is c.u.b. in δ. Let C2 = E2 ∩ F2. 0 0 0 Let C0 be the set of limit points of C0, and likewise for C1, C2. To ﬁnish, we show the following.

0 0 0 Claim 36. (C0,C1,C2) is homogeneous for P. 23 0 0 0 Proof. Suppose (F, G, H) is a block function from 3 × ϑ into (C0,C1,C2) of the correct type (since ϑ is countable, this just means that F , G, H are increasing, discontinuous, and have range in points of cofinality ω). ¯ ¯ Let F : ω · ϑ → κ be increasing and induce F , that is, F (i) = supj<ω·(i+1) F (j) ω for all i < ϑ. Let x ∈ ω be such that for all j < ω · ϑ,(x)π(j) ∈ P0 and ¯ ϕ0((x)π(j)) = F (j). ¯ + ¯ Let G: ω · ϑ → κ be increasing and induce G, that is G(i) = supj<ω·(i+1) G(j) for all i < ϑ. We may assume G¯ has range in C1. Since C1 ⊆ F1, for each j < ω · ϑ ω + we may get a yj ∈ ω (using countable choice) such that Tyj is wellfounded and ¯ + ω G(j) = [α 7→ |Tyj ¹α|]µ. Let y ∈ ω be such that for all j < ω·ϑ we have (y)π(j) = yj. Let H¯ : ω · ϑ → δ be increasing and induce H. We may assume H has range ω ++ in C2. Since C2 ⊆ F2, for each j < ω · ϑ there is a zj ∈ ω such that Tzj is ¯ ++ ω wellfounded and H(j) = [α 7→ [β 7→ |Tzj ¹β|]µα ]µ. Let z ∈ ω be such that for all j < ω · ϑ we have (z)π(j) = zj. Let (x0, y0, z0) = τ(x, y, x). Recall we identify ω · ϑ with lexicographic order on pairs (i, n) where i < ϑ and n ∈ ω. Since x ∈ W and τ is winning for Player II, 0 it follows that x ∈ W as well. Let F¯x0 : ω · ϑ → κ be the function determined by 0 ¯ 0 ¯ x , that is, Fx0 (j) = |(x )π(j)|. Since ran(Fx) ⊆ C0, it follows from the definition of C0 that F¯x0 (i, n) < F¯x(i, n + 1) < F¯(i) for all i < ϑ. So, for each i < ϑ we have ¯ ¯ supn max{Fx(i, n), Fx0 (i, n)} = F (i). Thus the function Fx,x0 jointly produced by x and x0 is equal to F . Consider next y and y0. For all j < ω · ϑ we have that T + and T ++ are (y)π(j) (z)π(j) ¯ wellfounded. Let α0 = supi<ϑ F (i) = supj<ω·ϑ F (j). Letg ¯y be the block function + determined by y. That is, for α ∈ C0 and j < ω · ϑ,g ¯y(α, j) = |T ¹α|. For (y)π(j) α α µ almost all α the functiong ¯y (j) =g ¯y(α, j) is increasing. Let gy be the function α α α α induced byg ¯y , that is, gy (i) = supn g¯y (j) where j = (i, n). For µ almost all α, gy α has range in the limit points of D (as defined above in the construction of C1). Say M1 ⊆ C0 is this measure one set. Consider any α ∈ M1 with α > α0. Then α α for any j < ω · ϑ we have that (x, y, z) ∈ Aα,η,j where η =g ¯y (j) < gy (i) (where α + α again j = (i, n)). It follows from the definition of D that |T 0 ¹α| < g (i) as (y )π(j) y α 0 α + α well. So, ifg ¯ 0 is the function determined by y (i.e.,g ¯ 0 (j) = |T 0 ¹α|), theng ¯ 0 y y (y )π(j) y α α andg ¯y both induce the function gy . It follows that the function Gy,y0 they jointly produce is equal to G. 0 + ¯α Consider finally z and z . For α ∈ M1, β < α , and j < ω · ϑ, let hz (β, j) = |T ++ ¹β|. Since H¯ is increasing and induces H, it follows from the definition of (z)π(j) 0 ∗ ∗ ¯α 0 ¯α z that if j < j then ∀µα ∀µα β hz (β, j ) < hz (β, j). This imolies that there is a µ 0 0 + measure one set M2 ⊆ M1 such that for α ∈ M2 there is a c.u.b. C ⊆ α which is ¯α ¯α closed under hz and such that the map (β, j) 7→ hz (β, j) is order-preserving when restricted to pairs with β ∈ C, cf(β) = ω. Also, from the definition of E2 there is 0 a µ measure one set M2 ⊆ M2 such that for α ∈ M2 we have that there is a c.u.b. C ⊆ α+ such that for β ∈ C with cf(β) = ω we have in addition that for all i < ϑ α α that supn hz (β, j) ∈ E2 , where j = (i, n). + Consider now α ∈ M2 with α > α0 (α0 as above). Fix a c.u.b. C ⊆ α as with the two properties specified immediately above. Let β ∈ C with cf(β) = ω α α and β > sup gy (j). For such β, if j < ω · ϑ and η = hz (β, j), then from the α definition of Aα,β,η,j we see that (x, y, z) ∈ Aα,β,η,j . From the definition of E2 it 24 α α 0 follows that hz0 (β, j) < sup hz (β, j ) where j = (i, n), and the supremum ranges 0 0 over j = (i, n ). It follows that the function H¯z0 induces the function H, that is, H = Hz,z0 . Since τ is winning for Player II, we have that P(F, G, H) = 1 and we are done. q.e.d. (Claim 36)

We have proved the claim, and this ﬁnishes the proof of Theorem 25 (and thus the proof of Theorem 24).

7. A polarized partition property with higher exponents We now improve Theorem 24 from countable exponents to arbitrary exponents ϑ < κ. The setup is the same as in the proof of Theorem 24: we have the (Steel) pointclass Γ ⊆ S(κ) forming the lowest level of the projective-like hierarchy con- ω taining S(κ) which is scaled, non-selfdual, closed under ∀ω and ﬁnite intersections and unions, and let ∆ = Γ ∩ Γ˘. Theorem 37. Assume AD. Let κ be a weakly inaccessible Suslin cardinal. Then for all ϑ < κ we have (κ, κ+, κ++) → (κ, κ+, κ++)ϑ.

Proof. We fix ϑ < κ, and fix a prewellordering ¹ ∈ ∆ of length ω · ϑ. We shall use the coding lemma to code functions from ω · ϑ to κ. We again identify the ordinals below ω · ϑ with the pairs (i, n) ordered lexicographically, where i < ϑ and n < ω. We shall also use the trees T + and T ++ from the proof of Theorem 24. ωω Fix a pointclass Γ0 ⊆ ∆ which is non-selfdual, is closed under ∃ , has the prewellordering property, and contains the prewellordering ¹. Fix a Γ0-universal set U. Without loss of generality we may assume dom(¹) = ωω. Let |a| denote the rank of a ∈ ωω in ¹. For x ∈ ωω, we say x codes a function at j < ω · ϑ provided (i) ∀a (|a| = j → ∃b U(x, a, b)) 0 0 0 0 0 0 (ii) ∀a, a , b, b (U(x, a, b) ∧ U(x, a , b ) ∧ |a| = |a | = j → (b, b ∈ P0 ∧ ϕ0(b) = 0 ϕ0(b )) We say x codes a function from ω · ϑ to κ (or just say x codes a function) if x codes a function at j for all j < ω · ϑ. Recall that if S is a tree, we write S(γ) to denote the subtree of S consisting of points in S below γ (we are identifying ordinals with finite tuples here for convenience). Fix now a partition P of the block functions from 3×ϑ to (κ, κ+, κ++). Consider the game G where player I plays out the real hx, y, u, z, v, wi and player II plays out the real hx0, y0, u0, z0, v0, w0i. Suppose first that there is a least j < ω · ϑ+ such that x or x0 does not code a function at j. In this case, player II wins iff x does not code a function at j. Suppose next that both x and x0 code functions. If there is a least j such that y or y0 does not code a function at j, player II again wins iff y does not code a function at j. Likewise, if all of x, y, x0, y0 code functions and there is a least j such that z or z0 doesn’t code a function at j, player II wins iff z doesn’t code a function at j. 0 0 0 ¯ ¯ Suppose next that x, y, z, x , y , z all code functions from ω·ϑ to κ. We let fx, fy, etc. denote these functions. Let α ∈ Cω be the least ordinal, if it exists, such that one of the following holds. + ¯ + ¯ (a) There is a j < ω · ϑ such that either Tu ¹α(fy(j)) or Tu0 ¹α(fy0 (j)) is illfounded. 25 + ¯ + ¯ (b) There is a j < ω · ϑ such that either Tv ¹α(fz(j)) or Tv0 ¹α(fz0 (j)) is illfounded. + ++ + ¯ (c) There is a β < α and a j < ω · ϑ such that either Tw ¹β(|Tv ¹α(fz(j))|) ++ + ¯ or Tw0 ¹β(|Tv0 ¹α(fz0 (j))|) is illfounded. Suppose first such an α exists. If (a) holds, let j be least such that either + ¯ + ¯ + ¯ Tu ¹α(fy(j)) or Tu0 ¹α(fy0 (j)) is illfounded. Player II then wins iff Tu ¹α(fy(j)) is illfounded. If (a) does not hold, but (b) holds, then let j be least such that either + ¯ + ¯ + ¯ Tv ¹α(fz(j)) or Tv0 ¹α(fz0 (j)) is illfounded. Player II then wins iff Tv ¹α(fz(j)) is illfounded. If (a) and (b) do not holds but (c) holds, then let (β, j) be the lexico- ++ + ¯ graphically least pair witnessing (c). Player II then wins iff Tw ¹β(Tv ¹α(fz(j))) is illfounded. Finally, if no such α exists, then let F = Fx,x0 be the function jointly produced ¯ ¯ ¯ ¯ ¯ by fx and fx0 . That is, F (i) = supn max{fx(i, n), fx0 (i, n)} for all i < ϑ. Letg ¯y,u be the block function defined as follows. For α ∈ Cω and j < ω · ϑ, let + 0 0 ¯ g¯y,u(α, j) = |Tu ¹α(fy(j))|. Likewise defineg ¯y0,u0 using y and u . Let Gy,u be the + function from ω · ϑ to κ represented byg ¯y,u. That is, G¯y,u(j) = [α 7→ g¯y,u(α, j)]µ. + Likewise define G¯y0,u0 . Finally, let G = Gy,u,y0,u0 : ϑ → κ be the function they ¯ ¯ jointly produce: G(i) = supn max{Gy,u(i, n), Gy0,u0 (i, n)}. ¯ + Let hz,v,w be the block function defined as follows. For α ∈ Cω, β < α , ¯ + + ¯ and j < ω · ϑ, let hz,v,w(α, β, j) = |Tw ¹β(|Tv ¹α(fz(j))|). Notice we may write ¯ + ¯ 0 0 this as hz,v,w(α, β, j) = |Tw ¹β(¯gz,v(α, j))|. Similarly define hz0,v0,w0 using z , v , 0 ++ ¯ w . Let H¯z,v,w : ω · ϑ → κ be the function represented by hz,v,w. That is, ++ ¯ ¯ 0 0 0 Hz,v,w(j) = [α 7→ [β 7→ hz,v,w(α, β, j)]µα ]µ. Let H = Hz,v,w,z ,v ,w : ϑ → κ be ¯ ¯ the function they jointly produce: H(i) = supn max{Hz,v,w(i, n), Hz0,v0,w0 (i, n)}. Player II then wins the run of the game G provided P(F, G, H) = 1. Suppose without loss of generality that player II has a winning strategy τ for G. 0 0 0 + ++ We define c.u.b. sets C0, C1, C2 in κ, κ , κ respectively which are homogeneous for P. The argument is similar to that of Theorem 24, so we shall concentrate on the differences. 0 We first define C0 (C0 will be the set of limit points of C0). For each η < κ and j < ω · ϑ, Let

0 0 0 Aη,j = {c = hx, y, u, z, v, wi ; ∀j ≤ j (x codes a function at j ∧ fx(j ) ≤ η)}.

Clearly Aη,i ∈ ∆. Since τ is winning for player II, if c ∈ Aη,i, and τ(c) = 0 0 0 0 0 0 0 0 hx , y , u , z , v , w i, then for all j ≤ j, fx0 codes a function at j . By boundedness, 0 0 0 0 0 0 0 ρ(η, j) := sup{fx0 (j ); hx , y , u , z , v i ∈ τ[Aη,j ] ∧ j ≤ j} < κ.

Let C0 ⊆ C be c.u.b. and closed under f. We next deﬁne C1. For η < κ, η ∈ C0, with cf(η) > ϑ (for convenience), α ∈ Cω, α > η, j < ω · ϑ, and δ < α+, let

¯ ¯ ¯ Aη,α,j,δ ={c = hx, y, u, z, vi ; x, y, z code functions ∧ (∀j fx(j), fy(j), fz(j) ≤ η) 0 0 + 0 ¯ 0 ∧ ∀(α , j ) ≤lex (α, j) |Tu ¹α (fy(j )| ≤ δ ∧ ∀α0 < α ∀β0 < (α0)+ ∀j0 < ω · ϑ ++ 0 + 0 ¯ 0 Tw ¹β (|Tv ¹α (fz(j ))|) is wellfounded}. 26 0 0 0 0 0 0 0 Let c = hx , y , u , z , v , w i = τ(c), where c = hx, y, u, z, v, wi ∈ Aη,α,j,δ. From the second and third conjuncts in the above deﬁnition it follows that the least α0 such that (a), (b), or (c) holds for c, c0 is at least α. Furthermore, the second con- 0 + 0 0 junct gives that (a) does not hold at α for all j ≤ j. It follows that Tu0 ¹α (fy0 (j )) 0 0 is wellfounded for all (α , j ) ≤lex (α, j). For η, α, j, δ as above, deﬁne

+ 0 0 0 0 0 0 0 0 ρ1(η, α, j, δ) = sup{Tu0 ¹α(fy0 (j )) ; j ≤ j ∧ c = hx , y , u , z , v i ∈ τ[Aη,α,j,δ]}.

+ Claim 38. ρ1(η, α, j, δ) < α .

α Proof. We define a Σ1 wellfounded relation of length at least ρ1(η, α, j, δ). Since α + Σ1 = S(α), it then follows that ρ1(η, α, j, δ) < α . In defining this relation we again for convenience think of the elements of T + as ordinals rather than tuples of ordinals. Define a relation S by letting

0 0 0 0 0 0 (b,u, s)S(b , u , s ) ↔ b = b ∧ u = u ∧ ∃c ∈ Aη,α,j,δ ∃y ∃j ≤ j ∃a 0 0 [τ(c)1 = y ∧ τ(c)2 = u ∧ |a| = j ∧ U(y, a, b) ∧ (b ∈ P0 ∧ ϕ0(b) ≤ η) ∧ s, s ∈ P0 0 0 ∧ ϕ (s), ϕ (s ) < α ∧ (ϕ (s) < + ϕ (s ) < + ϕ (b))] 0 0 0 Tu ¹α 0 Tu ¹α 0 + where < + refers to the Kleene-Brouwer ordering on the tree T ¹α. From the Tu ¹α u + α above remarks we have that S is wellfounded. From the coding lemma, T ¹α is Σ1 α α in the codes (relative to P0¹α). Also, Aη,α,j,δ ∈ ∆1 using the closure of ∆1 under + <α unions and intersections (for the last conjunct in the deﬁnition of Aη,α,j,δ note ++ 0 + 0 ¯ 0 0 + that if wellfounded |Tw ¹β (|Tv ¹α (fz(j ))|)| must have rank less that (α ) < α). α Since also η < α, it follows that S ∈ Σ1 , and we are done. q.e.d.

α + We let C1 be the set of ordinals below α closed under the ρ1 function. Let + α C1 ⊆ κ be the set of [G]µ where G(α) ∈ C1 for µ almost all α. The deﬁnition of C2 is similar to that of C1. For η < κ, η ∈ C0, with cf(η) > ϑ, + α ∈ Cω, α > η, j < ω · ϑ, and β, δ < α , let

¯ ¯ ¯ Aη,α,j,β,δ ={c = hx, y, u, z, v, wi ; x, y, z code functions ∧ (∀j fx(j), fy(j), fz(j) ≤ η) 0 0 + 0 ¯ 0 ∧ ∀α ≤ α ∀j < ω · ϑ |Tu ¹α (fy(j )| ≤ δ 0 0 + 0 ¯ 0 ∧ ∀α ≤ α ∀j < ω · ϑ |Tv ¹α (fz(j )| ≤ δ ∧ ∀α0 < α ∀β0 < (α0)+ ∀j0 < ω · ϑ ++ 0 + 0 ¯ 0 Tw ¹β (|Tv ¹α (fz(j ))|) is wellfounded 0 0 ++ 0 + ¯ 0 ∧ ∀(β , j ) ≤lex (β, j) |Tw ¹β (|Tv ¹α(fz(j ))|)| ≤ δ}. α 0 0 0 0 0 0 0 It again follows that Aη,α,j,β,δ ∈ ∆1 . Suppose c hx , y , u , z , v , w i = τ(c) where c = hx, y, u, z, v, wi ∈ Aη,α,j,β,δ. Since x, y, z all define functions taking values below 0 0 0 0 η, and since η ∈ C0 it follows that x , y , z also all code functions below η (for y , 0 ¯ ¯ z we use also that cf(η) > ϑ so that sup(fy), sup(fz) are also below η). From the second, third, and fourth conjuncts in the definition of Aη,α,j,β,δ and the fact that τ is winning for player II it follows that the least α0 such that (a), (b), or (c) holds at α0 is ≥ α. Also, from the second conjunct it follows that (a) cannot hold at α. From the third conjunct it follows that (b) cannot hold at α. From the last 27 0 + 0 0 0 conjunct it then follows that for all β < α and j with (β , j ) ≤lex (β, j) that ++ 0 + ¯ 0 Tw0 ¹β (|Tv0 ¹α(fz0 (j ))|) is wellfounded. For η, α, β, j, δ as in the definition of Aη,α,j,β,δ define

++ 0 + ¯ 0 0 0 ρ2(η, α, β, j, δ) = sup{|Tw0 ¹β (|Tv0 ¹α(fz0 (j ))|)| ;(β , j ) ≤lex (β, j) 0 0 0 0 0 0 ∧ c = hx , y , u , z , v i ∈ τ[Aη,α,β,j,δ]}. Analogous to Claim 38 we have:

+ Claim 39. ρ2(η, α.β.j, δ) < α .

Proof. The proof follows from a computation as in Claim 38, using the fact that |β| = α, so T ++¹β is isomorphic to a tree on α. q.e.d.

α + For α ∈ Cω, let now C2 be the c.u.b. subset of α consisting of points closed α ++ under the ρ2 function. Let D ⊆ α be the c.u.b. set of ordinals of the form [h]µα α ++ where ran(h) ⊆ D2 . Finally, let C2 ⊆ κ the the c.u.b. set of ordinals of the form α [H]µ where H(α) ∈ D for µ almost all α. 0 0 0 We now show that the c.u.b. sets C0, C1, C2 are homogenous for the given 0 0 0 partition P. Fix a block function (F, G, H) from 3×ϑ to (C0,C1,C2) of the correct type. Let F¯, G¯, H¯ from 3×ω ·ϑ to (κ, κ+, κ++) be increasing and induce (F, G, H). From the coding lemma, let x be such that x codes a function at j for all j < ω·ϑ ¯ and such that fx = F¯. + + + Since κ is regular by Theorem 24, sup(G) < κ . Let u be such that Tu is + wellfounded and [α 7→ |Tu ¹α|]µ > sup(G). Letg ˜: ω · ϑ → κ be defined as follows. ∗ For j < ω·ϑ, letg ¯(j) be the ordinal less than κ such that ∀µα [α 7→ |Tu¹α(¯g(j))|]µ = G¯(j). This is well-defined by the normality of µ and the definition of u. Let y be such that y codes the functiong ˜ (i.e., for all j < ω · ϑ, y codes a function at j, and the value coded at j isg ˜(j)). Since κ++ is also regular by Theorem 24, sup(H) < κ++. From the previous ++ section there is a real w such that Tw is wellfounded and such that [α 7→ [β 7→ ++ + |Tw ¹β]µα ]µ > sup(H). Define a function `: ω · ϑ → κ as follows. For j < ω · ϑ, let `(j) < κ+ be the ordinal represented by α 7→ `(j, α) with respect to µ, where for almost all α ∈ Cω we have: ++ H(j) = [α 7→ [β 7→ |Tw ¹β(`(j, α))|]µα ]µ.

This is well-defined from the definition of w and the fact that each µα is normal. Let z, v be the reals corresponding to ` just as y, u correspond tog ¯. So, z codes a ¯ + function fz from ω · ϑ to κ, and Tv is wellfounded. Consider the run of the game where player I plays c = hx, y, u, z, v, wi, and player II responds with c0 = τ(c) = hx0, y0, u0, z0, v0, w0i. Let η be the least point in C0 greater than max{sup(fx), sup(fy), sup(fz)} which has cofinality greater than ϑ. ¯ 0 ¯ Since fx = F¯ has range in C0, it follows that x also codes a function fx0 : ω ·ϑ → κ, and the first function they jointly produce, namely, ¯ ¯ Fx.x0 (i) = sup max{fx(i, n), fx0 (i, n)} n is equal to F . Consider now α > η in Cω. For such α and any j = (i, n) < ω · ϑ, let δ = + ¯ + |Tu ¹α(fy(j)| < α . An easy argument shows, as in the proof of the last section, 28 + ¯ that there is a µ measure one set of α such that the map j 7→ |Tu ¹α(fy(j)| is increasing, and we may assume α is in this set. By definition we have c ∈ Aη,α,j,δ. 0 + ¯ + ¯ Thus, c ∈ τ[Aη,α,j,δ]. Hence Tu0 ¹α(fy0 (j)) is wellfounded and |Tu0 ¹α(fy0 (j))| ≤ ρ1(η, α, j, δ). α Claim 40. For µ almost all α and all i < ϑ, supn g¯y,u(i, n) ∈ C1 .

0 Proof. We have ran(G) ⊆ C1. If for almost all α there is an i < ϑ for which this fails, them by the κ-completeness of µ we could ﬁx a i which witnessed the failure ¯ for almost all α. Now, G(i) = supn G(i, n) is represented with respect to µ by the function α 7→ supn g¯y,u(α, (i, n)). α 0 So, for µ almost all α, supn g¯y,uu(α, (i, n)) ∈ (C1 ) . q.e.d.

It follows that ρ1(η, α, j, δ) < supn g¯y,u(α, (i, n)) = gy,u(α, i), where j = (i, m) for some m. Thus, for µ almost all α we have that for all i < ϑ that

sup max{g¯y,u(α, (i, n)), g¯y0,u0 (α, (i, n))} = gy,u(α, i). n

It follows that the second function Gy,u,y0,u0 the players jointly produce is equal to G. By a similar argument, the third function Hz,v,w,z0,v0,w0 they jointly produce is + equal to H. Here we consider α > η and β < α such that β > supj g¯y,u(α, j), and β > supj g¯z,v(α, j). For such α, β we have that c ∈ Aη,α,j,β,δ where δ = ¯ hz,v,w(α, β, j). We assume here that α is in the µ measure one set such that the ¯ function (β, j) 7→ hz,v,w(α, β, j) is order-preserving when restricted to a c.u.b. C ⊆ α+ (as in the proof of the previous section). An easy argument as above shows that we may assume that for µ almost all α, and for µα almost all β, and all i < ϑ that α ∗ ∗ ¯ ¯ 0 0 0 supn hz,v,w(α, β, (i.n)) ∈ C2 . Thus, ∀µα ∀µα β ∀i < ϑ (supn hz ,v ,w (α, β, (i, n)) = ¯ supn hz,v,w(α, .β, (i, n))). It follows that Hz,v,w,z0,v0,w0 = H. Since τ is winning for Player II it follows that P(F, G, H) = 1, and we are done. q.e.d.

8. The strong polarized partition property In this section, we now prove the optimal result which was used in the applications in §5. Theorem 41. Assume AD. Let κ be a weakly inaccessible Suslin cardinal. Then (κ, κ+, κ++) → (κ, κ+, κ++)κ. The proof of Theorem 41 is again similar to that of the previous results, and we again use the trees T + and T ++ from before. Let P denote the given partition of the block functions of the correct type from (κ, κ, κ) to (κ, κ+, κ++). We now code functions from κ to κ using the uniform coding lemma (cf. ω ω [KKMW81]). Let U ⊆ ω × ω be universal for the syntactic class Σ1(Q) where 0 Q is a binary predicate symbol. Recall A ∈ Σ1(Q) if A(x) ↔ A (x0, x) ↔ 1 ∃y (B(x, y) ∧ ∀n Q((y)n)) where B ∈ Σ1. So, we may define the universal set 1 U by: Uz(x, y) ↔ ∃w (S(z, hx, yi, w) ∧ ∀n Q((w)n)), where S is universal for Σ1. Recall P is our Γ-complete set, and {ϕm} a Γ-scale on P (with norms onto κ). Let Pα = {x ∈ P ; |x| = α} be the set of codes for α < κ. Also, R is the prewellordering on P given by ϕ0, so R ∈ Γ. For α < κ, recall also Rα = {(x, y) ∈ 29 0 R ; ϕ0(y) < α} is the restriction of R to reals of norm less than α. Let Rα be the restriction of R to reals of norm ≤ α. 0 Let {ρn} be a Γ-scale on R, and let ρα (or ρα) denote its restriction to Rα (or 0 0 Rα). For any α < κ, ρα is a ∆-scale on Rα (similarly for ρα). Uniformly in α, 0 0 the scale ρα induces a scale on U(Rα). This gives, uniformly in α, a uniformizing 0 0 0 relation U(Rα) ∈ ∆ (uniformizing on the last coordinate) of U(Rα). In fact, U(Rα) 0 is in the projective hierarchy containing Rα. For α < κ, let α0 < κ be the least reliable ordinal ≥ α (with respect to the ω ω scale {ϕn} on P ). We let G: κ → ω be the Lipschitz continuous generic coding function from the Kechris-Woodin theory of generic codes for uncountable ordinals (cf. [KW08] for the theory of generic codes). This means G has the following properties. For all s ∈ κω, G(s) ∈ P . Also, for all α < κ, and any s ∈ (α0)ω enumerating an honest set S ⊆ α0, |G(αas)| = α. Here (and throughout this section) |z| denotes ϕ0(z). For α < κ, we say that comeager many x ∈ Pα have ω ∗ property B (where B ⊆ ω ), written ∀ x ∈ Pα B(x), if player II has a winning 0 <ω strategy in the game where players I and II play si ∈ (α ) and player II wins the a a a run iff G(α s0 s1 ··· ) ∈ B. If B is Suslin and co-Suslin, then this game is Suslin and co-Suslin as well, and hence determined (cf. [KKMW81, Theorem 2.5]). We also write ∀∗s ∈ (α0)ω to denote that player II has a winning strategy in the game 0 <ω a a where I and II play si ∈ (α ) to produce s = s0 s1 s2 ··· . We code functions from κ to κ using the uniform coding lemma as follows. For any f : κ → κ there is a real x such that for all α < κ, the set Ux(Rα) codes f¹α. That is, Ux(Rα)(a, b) iff ϕ0(a) < α, b ∈ P , and ϕ0(b) = f(ϕ0(a)). We say x codes 0 a function at α if Ux(Rα) satisfies:

0 (i) For all a, |a| = α implies that there is some b with Ux(Rα)(a, b). 0 0 0 0 0 0 (ii) For all a, a , b, and b , we have that (Ux(Rα)(a, b) ∧ Uz(Rα)(a , b ) ∧ |a| = |a0| = α → |b| = |b0|) holds.

+ ω We code functions from κ to κ as follows. Given y ∈ ω , and given δ < α ∈ C0, we say y is good at (δ, α) if for comeager many a ∈ Pδ, there is a (unique) b such 0 + that U y(Rδ)(a, b), and for this b we have that Tb ¹α is wellfounded. We let gy(δ, α) + be the least ordinal <α such that for comeager many a ∈ Pδ, and b as above, + + |Tb ¹α| ≤ gy(δ, α). This is welldeﬁned using the fact that cf(α ) > ω and the additivity of category. We say y is good at α ∈ Cω if y is good at (δ, α) for all δ < α. If y is good + at α for µ almost all α ∈ Cω, then y codes the function Gy : κ → κ deﬁned by Gy(δ) = [α 7→ gy(δ, α)]µ. ++ ω We code functions from κ to κ as follows. Given z ∈ ω , δ < α ∈ Cω and + 0 β < α , we say z is good at (δ, α, β) if for for comeager many a ∈ Pδ, if U z(Rδ)(a, b), ++ + then Tb ¹β is wellfounded. We let hz(δ, α, β) be the least ordinal < α such that ++ for comeager many a ∈ Pδ, and b as above, |Tb ¹β| ≤ hz(δ, α, β). We say z is good at (α, β) if for all δ < α we have that z is good at (δ, α, β). We say z is good at α if + for all δ < α and all β < α , z is good at (δ, α, β). If for µ almost all α ∈ Cω and + µα almost all β < α we have that z is good at (α, β), then z codes the function ++ Hz : κ → κ given by

Hz(δ) = [α 7→ [β 7→ gz(δ, α, β)]µα ]µ. 30 Consider the game G where player I plays out reals (x, y, z) and player II plays out (x0, y0, z0). Let α < κ be the least ordinal, if one exists, such that one of the following holds. (1) For some δ < α we have that y or y0 is not good (δ, α). (2) For some δ < α and β < α+ we have that z or z0 is not good at (δ, α, β). (3) x or x0 does not code an ordinal at α. Suppose first that an α < κ satisfying (1), (2), or (3) exists, and let α be the least such. First we check to see if case (1) holds at α. If so, then player II wins the run iff for the least δ as in (1) we have that y is not good at (δ, α). Suppose next that case (1) does not hold at α. Then we check case to see if case (2) holds at α. If so, and if (β, δ) is the lexicographically least pair as in (2), then player II wins the run iff z is not good at (δ, α, β). Suppose next that (1) and (2) do not hold at α, but case (3) holds. Player II then wins provided x does not code an ordinal at α. Finally, suppose that there is no α < κ satisfying (1), (2), or (3). So, x, x0, both code functions fx, fx0 from κ to κ. Let F : κ → κ be defined from fx and fx0 0 as usual, that is, F (β) = supj<ω·(β+1) max{fx(j), fx0 (j)}. Similarly, y and y code + + functions Gy, Gy0 from κ to κ . These determine the function G: κ → κ in the 0 ++ usual way. Likewise, z and z determine Hz, Hz0 : κ → κ which then determine H : κ → κ++. Player II then wins the run of the game iff P(F, G, H) = 1. Suppose without loss of generality that player II has a winning strategy τ for the game, and we define + ++ homogeneous sets C0 ⊆ κ, C1 ⊆ κ , and C2 ⊆ κ . For η1, η2 < κ, let A(η1, η2) be the set of (x, y, z) satisfying the following:

(a) y is good at α for all α ≤ η1. (b) z is good at α for all α ≤ η1. (c) x codes a function at all α ≤ η1 and fx(α) ≤ η2. A straightforward computation using the closure of ∆ under quantiﬁers shows 0 0 0 that A(η1, η2) ∈ ∆. From the deﬁnition of G, if (x, y, z) ∈ A(η1, η2) and (x , y , z ) = 0 τ(x, y, z), then x codes a function at all α ≤ η1. By boundedness (since Γ is closed under ∧, ∨), it follows that

0 0 0 ρ0(η1, η2) := sup{fx0 (α); α ≤ η1 ∧ (x , y , z ) ∈ τ[A(η1, η2)]} < κ.

Let C0 ⊆ κ be a c.u.b. subset closed under ρ0. + For α ∈ Cω, δ < α, and η < α , Let A(δ, α, η) be the set of (x, y, z) satisfying: (a) y and z good at all α0 < α, and for all α0 < α x codes a function at α0 0 with fx(α ) < α. 0 0 0 (b) For all δ ≤ δ, y is good at (δ , α) and gy(δ , α) ≤ η.

α 0 ∗ 0 ω Lemma 42. For δ < α ∈ Cω, if X(x, y) ∈ Σ1 , then X (x) ↔ ∀ s ∈ (δ ) X(G(s), y) α is also in Σ1 .

α Proof. Write X(x, y) ↔ ∃z Y (x, y, z) where Y ∈ Π0 . Fix a non-selfdual pointclass ωω Γ0 closed under ∃ , ∧, ∨ contained within <α-Suslin and which has prewellorderings of length at least δ0 (the least reliable ≥ δ). Using the coding lemma, we code strategies on δ0 by reals. We then have: 31 0 0<ω ω 0<ω ω ω ω z ∈ X ↔ ∃w[w codes a strategy τw :(δ ) → (δ ) × ω × ω

∧ ∀(s, y, z) a run according to τw, Y (G(s), y, z)]

Saying w codes a strategy is projective over Γ0, as is coding a run according to α 0 α τw. Since D ∈ Π0 , it follows that X is Σ1 . q.e.d.

α Claim 43. A(δ, α, η) ∈ ∆1 .

+ α α + Proof. The set B = {w ; |Tw ¹α| ≤ η} is in ∆1 . Since ∆1 is closed under <α unions and intersections (Theorem 15), it is enough to show that A0 = A0(δ, α, η) α is ∆1 , where 0 ∗ 0 + z ∈ A ↔ ∀ a ∈ Pδ ∃b (U z(Rδ)(a, b) ∧ |Tb ¹α| ≤ δ). 0 α It is enough, by a symmetrical argument, to show that A ∈ Σ1 . This follows immediately from Lemma 42. q.e.d.

If y ∈ A(δ, α, η) and y0 = τ(y), then y0 is good at (δ, α). This follows from the winning conditions for player II in the game G, speciﬁcally the fact that case (1) is considered at stage α ﬁrst. 0 + Claim 44. sup{gy0 (α, δ); y ∈ τ[A(δ, α.η)]} < α . Proof. The supremum in question has length bounded by the length of the following ordering:

0 <ω ∗ 0 ω ∗ 0 ω y1 ≺ y2 ↔ (y1, y2 ∈ τ[A(δ, α, η)]) ∧ ∃s0 ∈ (δ ) ∀ s1 ∈ (δ ) ∀ s2 ∈ (δ ) ∃b , b (U (R0 )(G(s), b ) ∧ (U (R0 )(G(sas), b ) ∧ |T +¹α| < |T +¹α|)) 1 2 y1 δ 1 y2 δ 0 2 b1 b2 α α It follows from Lemma 42 that ¹ ∈ Σ1 provided we show that there is a Σ1 relation S(b , b ) which when restricted to pairs such that T +¹α and T +¹α are 1 2 b1 b2 wellfounded correctly computes the relation |T +¹α| ≤ |T +¹α|. To see this, let Γ b1 b2 n ω be a sequence of non-selfdual pointclasses closed under ∃ω , ∧, ∨ of Wadge ranks cofinal in α. Let Un be universal sets for Γn. Let ψn be a Γn prewellordering of length αn where supn αn = α. Each real z codes the relation Rz ⊆ α × α given S n n n by Rz = n Rz where Rz is the relation on αn defined by (α, β) ∈ Rz iff there are u and v such that ψn(u) = α, ψn(v) = β, and Un((z)n, u, v). From the coding lemma, every relation on α is coded in this manner by some z. We can then say that S(b , b ) holds iff there is a z such that R is an order-preserving map from T +¹α to 1 2 z b1 + α Tb ¹α. Using the closure of ∆1 under <α unions and intersections (Theorem 15), 1 α it is straightforward to verify that S ∈ Σ1 . q.e.d. (Claim 44)

Let now 0 ρ1(δ, α, η) = sup{gy0 (α, δ); y ∈ τ[A(δ, α.η)]}. 0 + 0 Let C1(α) be the c.u.b. subset of α of points closed under ρ1. Let C1 = [α 7→ 0 0 + 00 + C1(α)]µ, so C1 is a c.u.b. subset of κ . Let C1 be a c.u.b. subset of κ so that 00 ω between any two elements ρ1 = [f]µ < [g]µ = ρ2 of C1 , there is a b ∈ ω such that + ∗ + 0 00 Tb is wellfounded and ∀µα f(α) < |Tb ¹α| < g(α). Let C1 = C1 ∩ C1 . 32 ++ + Lastly, we define C2 ⊆ κ . For δ < α ∈ Cω, and β < η < α , let A(δ, α, β, η) be the set of (x, y, z) satisfying: (a) y and z good at all α0 < α, and for all α0 < α x codes a function at α0 0 with fx(α ) < α. 0 0 0 (b) y is good at (δ , α) for all δ < α, and gy(δ , α) < β. 0 0 0 0 0 0 (c) For all (β , δ ) ≤lex (β, δ), z is good at (β , δ ) and hz(δ , α, β ) ≤ η. α A computation as in the proof of Claim 43 shows that A(δ, α, β, η) ∈ ∆1 . If (x, y, z) ∈ A(δ, α, β, η) and (x0, y0, z0) = τ(x, y, z), then from the winning conditions on G it follows that z0 is good at (δ, α, β). A computation as in Claim 44 shows that 0 + ρ2(δ, α, β, η) := sup{hz0 (α, δ, β); z ∈ τ[A(δ, α.β, η)]} < α 0 + 0 ++ Let C2(α) be c.u.b. in α and closed under ρ2. Let C2 be those ρ < κ such that 0 ρ = [α 7→ [β 7→ `(α, β)]µα ]µ where `(α, β) ∈ C2(α). An easy argument shows that 0 ++ 00 ++ C2 is c.u.b. in κ . Let C2 be c.u.b. in κ such that between any two ordinals 00 ++ ρ1 < ρ2 of C2 , there is a z such that Tz is wellfounded and ρ1 < [α 7→ [β 7→ ++ 00 |Tz ¹β| < ρ2. From the proof of Claim 34 it follows that such a C2 exists. Let 0 00 C2 = C2 ∩ C2 . Suppose now that (F, G, H) are block functions of the correct type into the block c.u.b. sets (C0,C1,C2), and we show that P(F, G, H) = 1. Let F¯ : ω · κ → C0 induce F , and let x code the function F¯, that is, x is good at all α < κ and fx = F¯. Let G¯ : ω · κ → C1 induce G. There is a functiong ¯ which induces G¯ in the + following sense.g ¯(δ, α) is defined for all δ < α ∈ Cω, andg ¯(δ, α) < α . Also, G¯(δ) = [α 7→ g¯(δ, α)]µ for all δ < κ. From the normality of µ, we may assume 0 without loss of generality thatg ¯(δ, α) ∈ C1(α) for all α ∈ Cω and that for all α ∈ Cω that δ 7→ g¯(δ, α) is strictly increasing. We code (some) c.u.b. subsets of κ by reals as follows. Say σ is a code if for all w ∈ P , σ(w) ∈ P . In this case, let Cσ be the c.u.b. subset of κ closed under σ, that is, Cσ = {α ; ∀w (w ∈ P<α → σ(w) ∈ P<α)}. An easy boundedness argument shows that Cσ is actually c.u.b. in κ. Also, an easy Solovay game argument shows that every c.u.b. C ⊆ κ contains a subset of the ¯ 00 + form Cσ. Since G has range in C1 , for each δ < κ there are reals w such that Tw ∗ + ¯0 is wellfounded and ∀µα g¯(δ, α) ≤ |Tw ¹α| < g¯(δ + 1, α). For each δ < κ, let G (δ) ¯ ¯ + be the least ordinal between G(δ) and G(δ + 1) which is of the form [α 7→ |Tw ¹α]µ + 0 0 for some w with Tw wellfounded. There is a functiong ¯ , withg ¯ (δ, α) defined for 0 0 δ < α ∈ Cω, such that for all δ < κ we have G¯ (δ) = [α 7→ g¯ (δ, α)]µ and for all α, δ 7→ g¯0(δ, α) is increasing. Also, we may assumeg ¯(δ, α) ≤ g¯0(δ, α) < g¯(δ + 1, α) for all δ < α ∈ Cω. From the uniform coding lemma, let y ∈ ωω be such that: 0 (1) For all δ < κ and all a ∈ Pδ, there is a (unique) hb, ci such that U y(Rδ)(a, hb, ci). 0 + (2) For all a ∈ Pδ and hb, ci such that U y(Rδ)(a, hb, ci), Tb is wellfounded and + ¯0 [α 7→ |Tb ¹α|]µ = G (δ). (3) For such a, b, c we have that Cc codes a c.u.b. subset of κ such that for all + 0 α ∈ Cc ∩ Cω we have that |Tb ¹α| =g ¯ (δ, α) (where δ = |a| as above). For δ < α ∈ Cω, let 0 `(δ, α) = sup{|c(x)| ; ∃a, b, x (a ∈ Pδ ∧ U y(Rδ)(a, hb, ci) ∧ x ∈ P<α}. By boundedness, `(δ, α) < κ. Let E ⊆ κ be a c.u.b. subset closed under `. For all 0 α ∈ E ∩ Cω we have that y is (δ, α) good and gy(δ, α) =g ¯ (δ, α) for all δ < α. 33 In a similar manner, we let H¯ : κ → κ++ induce H, and let h¯ be defined for + δ < α ∈ Cω and β < α and such that for all δ < ω · κ, H¯ (δ) = [α 7→ [β 7→ ¯ ¯ h(δ, α, β)]µα ]µ. We may assume that for all α < κ that (β, δ) 7→ h(δ, α, β) is ¯ 00 increasing (with respect to lexicographic order), and h(δ, α, β) ∈ C2 (α) for all + 0 δ < α ∈ Cω and β < α . For δ < ω · κ, let H¯ (δ) < H¯ (δ + 1) be least such that for ω ++ ¯ 0 ++ some w ∈ ω , Tw is wellfounded and H (δ) = [α 7→ [β 7→ [|Tw ¹β|]µα ]µ. An easy ¯0 + argument shows that there is a h such that for all δ < α ∈ Cω and β < α we have h¯(δ, α, β) < h¯0(δ, α, β) < h¯(δ + 1, α, β), and for each δ there is a real w such ++ ∗ ∗ ¯0 ++ that Tw is wellfounded and ∀µα ∀µα β h (δ, α, β) = |Tw ¹β|. We say a pair (σ, w) ++ codes a measure one set if σ codes a c.u.b. subset Cσ of κ (as above) and Tw is ++ wellfounded. We let A(σ,w) = {(α, β); α ∈ Cσ ∧ ∀γ < β (|Tw ¹γ| < β)}. Using the ++ ∗ tree T it follows that if A has measure one in the sense that ∀µα ∀µα β (α, β) ∈ A, then there is a (σ, w) with A(σ,w) ⊆ A. From the uniform coding lemma, fix a real z such that:

(1) For all δ < ω · κ and all a ∈ Pδ, there is a (unique) hb, c, di such that 0 U z(Rδ)(a, hb, c, di). 0 ++ (2) For all a ∈ Pδ and hb, c, di such that U z(Rδ)(a, hb, c, di), Tb is wellfounded ++ ¯ 0 and [α 7→ [β 7→ |Tb ¹β|]µα ]µ = H (δ). (3) For such a, b, c, d we have that (c, d) codes a measure one set A(c,d) such + ¯0 that for all (α, β) ∈ A(c,d) we have that |Tb ¹β| = h (δ, α, β) (where again δ = |a| ).

For δ < α ∈ D0 deﬁne: 0 `1(δ, α) = sup{|c(x)| ; ∃a, b, d (a ∈ Pδ ∧ U z(Rδ)(a, hb, c, di) ∧ x ∈ P<α}. + For δ < α ∈ D0 and β < α deﬁne: ++ 0 `2(δ, α, β) = sup{|Td ¹η| ; ∃a, b, c (a ∈ Pδ ∧ U z(Rδ)(a, hb, c, di) ∧ η < β}.

A boundedness argument as before shows that `1(δ, α) < κ. Likewise, a tree + argument shows that `2(δ, α, β) < α . Let D ⊆ κ be the c.u.b. set of points closed under `1. For α ∈ D ∩ Cω, let + Eα be the c.u.b. subset of α closed under `2. So, if α ∈ D ∩ Cω and β ∈ Eα, then (α, β) ∈ A(c,d) for all (c, d) such that for some a ∈ Pδ, δ < α, we have 0 U z(Rδ)(a, hb, c, di). Consider now the run of the game G where player I plays out (x, y, z) and player II responds with τ(x, y, z) = (x0, y0, z0). First note that x, y, and z are all 0 fully good (with the obvious meaning). In particular x codes a function fx0 : κ → κ. For any α < ω · κ,(x, y, z) ∈ A(α, F¯(α)). Also, A(α, F¯(α)) ∈ ∆. So, (x0, y0, z0) ∈ τ[A(α, F¯(α)] and therefore fx0 (α) < ρ0((α), F¯(α)) < F¯(α + 1) since F¯ has range in C0 which is closed under ρ0. So, the function Fx,x0 jointly produced from fx and fx0 is equal to F . Let Ey be the c.u.b. subset of κ as deﬁned above, the set of closure points of 0 0 ` = `y. Letg ¯,g ¯ be as in the deﬁnition of y, sog ¯(δ, α) ≤ g¯ (δ, α) < g¯(δ + 1, α) for all δ < α ∈ Cω. Let also gy be the function coded by y, since y is good. + That is, gy(δ, α) is the least γ < α such that for comeager many a ∈ Pδ, if 0 + U y(Rδ)(a, hb, ci), then |Tb ¹α| = gy(δ, α). So, for all α ∈ Ey ∩ Cω and δ < α 0 we have gy(δ, α) =g ¯ (δ, α). If α is in addition closed under the function F , then 0 we have that (x, y, z) ∈ A(δ, α, g¯ (δ, α)). For such δ, α it follows that gy0 (δ, α) < 0 `(δ, α, g¯ (δ, α)) < g¯(δ+1, α). So, for all δ < ω·κ, G¯y(δ) = [α 7→ gy(δ, α)]µ and Gy0 (δ) 34 upper bound lower bound Base Case #1: 1 [ M / M / M ] ZF + AD ZFC + WC Base Case #2: 2 [ M / M / ℵ3 ] ZFC + SC+M ZFC + WC Base Case #3: 3 [ M / M / ℵ2 ] ZF + AD ZFC + WC Base Case #4: 4 [ M / M / ℵ1 ] ZF + AD ZFC + WC (#1) 5 [ M / M / ℵ0 ] ZF + AD ZFC + WC Base Case #5a: 6 [ M / ℵ2 / M ] ZFC + 2MC ZFC + 2MC Base Case #5b: 7 [ M / ℵ2 / ℵ3 ] ZFC + MC ZFC + MC Base Case #5c: 8 [ M / ℵ2 / ℵ2 ] ZFC + MC ZFC + MC Base Case #5d: 9 [ M / ℵ2 / ℵ1 ] ZFC + MC ZFC + MC (#5a) 10 [ M / ℵ2 / ℵ0 ] ZFC + MC ZFC + MC Base Case #6: 11 [ M / ℵ1 / M ] ZF + AD ZFC + WC Base Case #7: 12 [ M / ℵ1 / ℵ3 ] ZFC + SC ZFC + WC 13 [ M / ℵ1 / ℵ2 ] 0 = 1 0 = 1 Base Case #8: 14 [ M / ℵ1 / ℵ1 ] ZF + AD ZFC + WC (#6) 15 [ M / ℵ1 / ℵ0 ] ZF + AD ZFC + WC (#1) 16 [ M / ℵ0 / M ] ZF + AD ZFC + WC (#2) 17 [ M / ℵ0 / ℵ3 ] ZFC + SC+M ZFC + WC 18 [ M / ℵ0 / ℵ2 ] 0 = 1 0 = 1 (#4) 19 [ M / ℵ0 / ℵ1 ] ZF + AD ZFC + WC (#1,#3) 20 [ M / ℵ0 / ℵ0 ] ZF + AD ZFC + WC (#1) 21 [ ℵ1 / M / M ] ZF + AD ZFC + WC (#2) 22 [ ℵ1 / M / ℵ3 ] ZFC + MC ZFC + MC (#3) 23 [ ℵ1 / M / ℵ2 ] ZF + AD ZFC + WC (#4) 24 [ ℵ1 / M / ℵ1 ] ZF + AD ZFC + WC (#1) 25 [ ℵ1 / M / ℵ0 ] ZF + AD ZFC + WC (#5a) 26 [ ℵ1 / ℵ2 / M ] ZFC + MC ZFC + MC (#5b) 27 [ ℵ1 / ℵ2 / ℵ3 ] ZFC ZFC (#5c) 28 [ ℵ1 / ℵ2 / ℵ2 ] ZFC ZFC (#5d) 29 [ ℵ1 / ℵ2 / ℵ1 ] ZFC ZFC (#5a) 30 [ ℵ1 / ℵ2 / ℵ0 ] ZFC ZFC

Figure 1. Lower and upper bounds for the consistency strength of patterns 1 to 30.

¯ ¯ 0 ¯ 0 are both less than Gy(δ + 1, α). So, supδ0<ω·(δ+1) max{Gy(δ ), Gy0 (δ )} = G(δ). So, the function jointly produced by y and y0 is equal to G. The argument for z, z0 is similar. Recall H : κ → κ++, H¯ : ω · κ → κ++, and ¯ ¯ ¯ ¯ ¯0 h(δ, α, β) induces H, that is, H(δ) = [α 7→ [β 7→ h(δ, α, β)]µα ]µ. Also, h is fixed ¯ ¯0 ¯ and h(δ, α, β) ≤ h (δ, α, β) < h(δ + 1, α, β). Let Dz ⊆ κ be the c.u.b. set of points α + closed under `1 as above. For α ∈ Dz ∩ Cω, let Ez ⊆ α be the c.u.b. set of points closed under `2 (more precisely, the function (δ, β) 7→ `2(δ, α, β)). Consider (α, β) α 0 such that α ∈ Ey, α ∈ Dz ∩Cω, β ∈ Ez , α is closed under F , β > supδ<α{g¯ (δ, α)}, 0 0 and for all β < β and δ < α, hz(δ, α, β ) < β. This set of pairs A has measure one ∗ ∗ set with respect the iterated measure, that is, ∀µα ∀µα β (α, β) ∈ A. For (α, β) ∈ A, ¯0 ¯ 0 z is in the set A(δ, α, β, h (δ, α, β)) for all δ < α. Since h has its range in the C2(α), ¯ hz0 (δ, α, β) < h(δ + 1, α, β). Thus, for all δ < ω · κ, H¯z(δ) and H¯z0 (δ) are both less than H¯ (δ + 1). It follows that the function jointly produced by z and z0 is equal to H. Since τ is winning for player II, it follows that P(F, G, H) = 1, and we are done. 35 9. Summary Figures 1 and 2 list all of the sixty patterns of measurability and cofinality for the first three uncountable cardinals. In the first column, we list “Base Case #n” if a pattern is one of our base cases. We list numbers in parentheses to indicate in which of the diagrams of § 3 the pattern shows up (if at all: of course, the 13 inconsistent patterns do not show up in the diagrams). For the purpose of listing the upper and lower consistency strength bounds of our patterns, we define the following theories: ZFC + SC+M stands for ZFC together with the statement “There are κ < λ where κ is supercompact and λ is measurable”; ZFC + SC stands for ZFC together with the statement “There is a supercompact cardinal”; ZFC+MC stands for ZFC together with the statement “There is a measurable cardinal”; ZFC + 2MC stands for ZFC together with the statement “There are two measurable cardinals”; ZFC + WC stands for ZFC together with the statement “There is a Woodin cardinal”.

Upper bounds. Most of the upper bounds come directly from our consistency proofs in Theorems 21, 16, 18, 20, 3, 22, 17, and 23 (corresponding to the eight base cases, respectively) and the reduction diagrams as listed in § 3. In a few cases, the upper bound for the consistency strength obtained by our reduction diagrams is patently not optimal. In our table, we have given the optimal bounds and brieﬂy list these exceptional cases in the following: patterns 22 and 26 can be obtained from a measurable cardinal by symmetrically collapsing it to the desired cardinal. Patterns 27, 28, 29, 30, 47, 48, and 50 all share the feature that ℵ2 is regular but non-measurable and do not involve any measurable cardinals; consequently, the methods of Theorem 3 allow us to obtain them from ZFC. Patterns 32 and 37 only involve one singular cardinal, and can thus be obtained from ZFC by symmetrically collapsing a strong limit of the desired coﬁnality. Finally, pattern 46 is another application of the methods of Theorem 3 that only requires one measurable cardinal.

Lower bounds. There are a number of trivial lower bounds: any pattern involving a measurable or two measurables necessarily has ZFC+MC or ZFC+2MC as a lower bound (by the standard L[U] argument). For other lower bounds, our main tool is the following theorem: Theorem 45 (Schindler / Jensen-Steel). Suppose δ < δ+ are singular. Then there is an inner model with a Woodin cardinal.

Proof. [Sch99, Theorem 1] proved this claim under the additional assumption that there is some Ω > δ+ that is inaccessible and measurable in HOD. Schindler needed this assumption to build the core model. In the meantime, Jensen and Steel have eliminated this assumption from the construction of the core model (cf. [JS07a, JS07b]). q.e.d.

Theorem 45 allows us to deal immediately with those patterns that have two consecutive singular cardinals (patterns 14, 15, 19, 20, 34, 35, 39, 40, 56, 57, and 60) and get a lower bound of a Woodin cardinal. Patterns that involve κ and κ+ such that either both are measurable or one of them is measurable and the other is singular have to be transformed into those that have two consecutive singulars by Pˇr´ıkr´yforcing via Theorem 1. Recall that in this paper, we deﬁned κ to be 36 upper bound lower bound

(#6) 31 [ ℵ1 / ℵ1 / M ] ZF + AD ZFC + WC (#7) 32 [ ℵ1 / ℵ1 / ℵ3 ] ZFC ZFC 33 [ ℵ1 / ℵ1 / ℵ2 ] 0 = 1 0 = 1 (#8) 34 [ ℵ1 / ℵ1 / ℵ1 ] ZF + AD ZFC + WC (#6) 35 [ ℵ1 / ℵ1 / ℵ0 ] ZF + AD ZFC + WC (#1) 36 [ ℵ1 / ℵ0 / M ] ZF + AD ZFC + WC (#2) 37 [ ℵ1 / ℵ0 / ℵ3 ] ZFC ZFC 38 [ ℵ1 / ℵ0 / ℵ2 ] 0 = 1 0 = 1 (#4) 39 [ ℵ1 / ℵ0 / ℵ1 ] ZF + AD ZFC + WC (#1,#3) 40 [ ℵ1 / ℵ0 / ℵ0 ] ZF + AD ZFC + WC (#1) 41 [ ℵ0 / M / M ] ZF + AD ZFC + WC (#2) 42 [ ℵ0 / M / ℵ3 ] ZFC + SC+M ZFC + WC (#3) 43 [ ℵ0 / M / ℵ2 ] ZF + AD ZFC + WC 44 [ ℵ0 / M / ℵ1 ] 0 = 1 0 = 1 (#1,#4) 45 [ ℵ0 / M / ℵ0 ] ZF + AD ZFC + WC (#5a) 46 [ ℵ0 / ℵ2 / M ] ZFC + MC ZFC + MC (#5b) 47 [ ℵ0 / ℵ2 / ℵ3 ] ZFC ZFC (#5c) 48 [ ℵ0 / ℵ2 / ℵ2 ] ZFC ZFC 49 [ ℵ0 / ℵ2 / ℵ1 ] 0 = 1 0 = 1 (#5a,#5d) 50 [ ℵ0 / ℵ2 / ℵ0 ] ZFC ZFC 51 [ ℵ0 / ℵ1 / M ] 0 = 1 0 = 1 52 [ ℵ0 / ℵ1 / ℵ3 ] 0 = 1 0 = 1 53 [ ℵ0 / ℵ1 / ℵ2 ] 0 = 1 0 = 1 54 [ ℵ0 / ℵ1 / ℵ1 ] 0 = 1 0 = 1 55 [ ℵ0 / ℵ1 / ℵ0 ] 0 = 1 0 = 1 (#1,#6) 56 [ ℵ0 / ℵ0 / M ] ZF + AD ZFC + WC (#2,#7) 57 [ ℵ0 / ℵ0 / ℵ3 ] ZFC + SC ZFC + WC 58 [ ℵ0 / ℵ0 / ℵ2 ] 0 = 1 0 = 1 59 [ ℵ0 / ℵ0 / ℵ1 ] 0 = 1 0 = 1 (#1,#3,#4,#6,#8) 60 [ ℵ0 / ℵ0 / ℵ0 ] ZF + AD ZFC + WC

Figure 2. Lower and upper bounds for the consistency strength of patterns 31 to 60.

measurable if there is a normal κ-complete ultrafilter on κ. This choice of definition allows us to transform patterns 1, 2, 3, 4, 5, 11, 12, 16, 17, 21, 23, 24, 25, 31, 36, 41, 42, 43, and 45 into a pattern with two consecutive singulars and thus apply Theorem 45 to get a lower bound of a Woodin cardinal. If one insists on the ordinary definition of “κ is measurable” (i.e., “there is a κ-complete ultrafilter on κ”), then this route is not available in general. Working in the base theory ZF + DC, it is possible to construct a normal ultrafilter from a κ-complete one (cf. [Jec03, Theorem 10.20]), but without additional assumptions, we do not know how to derive more strength than a measurable out of, say, [ ℵ0 / M / ℵ3 ].

Open Questions. We end the paper by listing some remaining open questions. Six of the eight base cases can be obtained from a model of ZF+AD, but Base Case 37 #2 appears to need (assumptions on the order of) ZFC + SC+M and Base Case #7 appears to need (assumptions on the order of) ZFC + SC.4 Question 46. Is it possible to force Base Case #2 and Base Case #7 from ZF+AD (thus reducing the consistency strength upper bound)? The two mentioned patterns are among 30 (out of our 60) patterns for which the upper bound and the lower bound in consistency strength do not coincide. Question 47. Can we determine the precise consistency strength in the cases where upper and lower bounds do not coincide? There are other large cardinal properties that can be exhibited by small cardinals, + such as “κ is κ -supercompact” (under AD, ℵ1 exhibits this property (cf. [DPH78])). Let us add another label for this property to our patterns, resulting in 4×5×6 = 120 patterns.

Question 48. Which of the 120 patterns involving cofinalities ℵ0, ℵ1, ℵ2, ℵ3, measurability and κ+-supercompactness are consistent? Note that a 1975 result of Martin (cf. [DPH78, § 2] for details) about the κ+- supercompactness of κ under the assumption that both κ and κ+ carry a normal measure produces some nontrivial restrictions for Question 48. Now, after considering all measurability and cofinality patterns for the cardinals ℵ1, ℵ2, and ℵ3, one could ask what happens if the same question is posed for the first four uncountable cardinals. There are 3 × 4 × 5 × 6 = 360 such patterns for the first four uncountable cardinals. Question 49. Which of the 360 measurability and cofinality patterns for the first four uncountable cardinals are consistent? Of course, a complete answer to Question 49 would require (among other things) a solution of one of the big open questions of the field of large cardinals without the Axiom of Choice, viz. whether it is consistent to have four consecutive measurable cardinals. As a consequence, we do not expect an answer to Question 49 very soon. Slightly less ambitious would be to ask the same question not for four consecutive cardinals, but for a different selection of three consecutive cardinals, e.g., the cardinals ℵ2, ℵ3, and ℵ4. Here we would have 4 × 5 × 6 = 120 patterns. Question 50. Which of the 120 measurability and cofinality patterns for the cardinals ℵ2, ℵ3, and ℵ4 are consistent? However, most of the methods used in this paper to handle the case of the first three uncountable cardinals will not work in this setting. The main reason is that most of the proofs require symmetrically collapsing some large cardinal to be ℵ1. This collapse is canonically well-orderable, and thus at our disposal in the choice- free situation. The collapse of a cardinal to be ℵ2, however, is not canonically well-orderable; consequently, the obvious analogues of our proofs will not work in the setting of Question 50. At this point, it might be useful to mention that some of the patterns have alternative consistency proofs that are more likely to transfer to the situation of

4As the proof of [AH86, Theorem 1] shows, slightly weaker supercompactness hypotheses (which are still well beyond the consistency strength of AD) actually suffice to establish Base Case #2 and Base Case #7. 38 ℵ2, ℵ3, and ℵ4. We would like to give one example: if there is a strongly compact cardinal κ, it is possible to obtain the pattern [ ℵ1 / ℵ0 / ℵ1 ] by using strongly compact Pˇr´ıkrýforcing. Obviously, this proof is not optimal in terms of consistency strength (as we can get it from ZF + AD via Theorem 20). However, this proof lifts to give a consistency proof of the pattern “ℵ2 is regular but not measurable, ℵ3 has 5 cofinality ℵ0, and ℵ4 has cofinality ℵ2”.

References

[AH86] Arthur W. Apter and James M. Henle. Large cardinal structures below ℵω. The Journal of Symbolic Logic, 51(3):591–603, 1986. [AH91] Arthur W. Apter and James M. Henle. Relative consistency results via strong compactness. Fundamenta Mathematicae, 139(2):133–149, 1991. [AHJ00] Arthur W. Apter, James M. Henle, and Stephen C. Jackson. The calculus of partition sequences, changing cofinalities, and a question of Woodin. Transactions of the American Mathematical Society, 352(3):969–1003, 2000. [Apt96] Arthur W. Apter. AD and patterns of singular cardinals below Θ. The Journal of Symbolic Logic, 61(1):225–235, 1996. [DPH78] Carlos A. Di Prisco and James M. Henle. On the compactness of ℵ1 and ℵ2. Journal of Symbolic Logic, 43(3):394–401, 1978. [Hen83] James M. Henle. Magidor-like and Radin-like forcing. Annals of Pure and Applied Logic, 25(1):59–72, 1983. [Jac08] Stephen C. Jackson. Suslin cardinals, partition properties, homogeneity. Introduction to part II. In Alexander S. Kechris, Benedikt Löwe, and John R. Steel, editors, Games, Scales, and Suslin Cardinals. The Cabal Seminar, Volume I, volume 31 of Lecture Notes in Logic, pages 273–313. Cambridge University Press, 2008. [KLS08, pages 273–313]. [Jac09] Stephen C. Jackson. Structural consequences of AD. In Matthew Foreman and Akihiro Kanamori, editors, Handbook of Set Theory, volume III. Springer-Verlag, 2009. To appear. [Jec03] Thomas Jech. Set theory: The third millenium edition, revised and expanded. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. [JS07a] Ronald B. Jensen and John R. Steel. Note on getting almost-true K, 2007. Unpub- lished manuscript. [JS07b] Ronald B. Jensen and John R. Steel. Note on weak covering, 2007. Unpublished manuscript. [Kan94] Akihiro Kanamori. The higher infinite. Large cardinals in set theory from their be- ginnings. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994. [Kec78] Alexander S. Kechris. AD and projective ordinals. In Alexander S. Kechris and Yian- nis N. Moschovakis, editors, Proceedings of the Caltech-UCLA Logic Seminar 1976– 77, volume 689 of Lecture Notes in Mathematics, pages 91–132. Springer-Verlag, Berlin, 1978.

5A sketch of the proof is as follows. Let κ < λ be such that in our ground model V, κ is strongly compact and λ is the least singular strong limit cardinal of cofinality ℵ2 greater than κ. Force over V with P1 ×P2, where P1 = Col(ℵ2, <κ) and P2 is strongly compact Pˇr´ıkrýforcing based on κ and λ as defined in the proof of [AH91, Theorem 1]. Let G = G1 × G2 be the resulting generic, with r = hrn ; n < ωi the generic sequence through Pκ(λ) generated by G2. For δ ∈ (κ, λ) a cardinal, define r¹δ = hrn ∩ δ ; n < ωi. Consider the symmetric model N := HDV({G1¹δ ; δ ∈ (ℵ2, κ) and δ is a cardinal} ∪ {r¹δ ; δ ∈ (κ, λ) and δ is a cardinal}). The arguments found in the proofs of [AH91, Theorem 1] and Theorems 2 and 3 of this paper then show that N is as desired. Note that if P1 is redefined as Col(ℵ1, <κ), λ is redefined as the least singular strong limit cardinal of cofinality ℵ1 greater than κ, P2 remains strongly compact Pˇr´ıkrýforcing based on κ and λ, and N is redefined as N := HDV({G1¹δ ; δ ∈ (ℵ1, κ) and δ is a cardinal} ∪ {r¹δ ; δ ∈ (κ, λ) and δ is a cardinal}), then N is a model of the pattern [ ℵ1 / ℵ0 / ℵ1 ]. 39 [KKMW81] Alexander S. Kechris, Eugene M. Kleinberg, Yiannis N. Moschovakis, and W. Hugh Woodin. The axiom of determinacy, strong partition properties and nonsingular measures. In Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis, editors, Proceedings of the Caltech-UCLA Logic Seminar, 1977–79, volume 839 of Lecture Notes in Mathematics, pages 75–99. Springer-Verlag, Berlin, 1981. Reprinted in [KLS08, pages 333–354]. [Kle70] Eugene M. Kleinberg. Strong partition properties for infinite cardinals. The Journal of Symbolic Logic, 35:410–428, 1970. [KLS08] Alexander S. Kechris, Benedikt Löwe, and John R. Steel, editors. Games, Scales, and Suslin Cardinals. The Cabal Seminar, Volume I, volume 31 of Lecture Notes in Logic. Cambridge University Press, 2008. [KSS81] Alexander S. Kechris, Robert M. Solovay, and John R. Steel. The axiom of determinacy and the prewellordering property. In Alexander S. Kechris, Donald A. Mar- tin, and Yiannis N. Moschovakis, editors, Proceedings of the Caltech-UCLA Logic Seminar, 1977–79, volume 839 of Lecture Notes in Mathematics, pages 101–125. Springer-Verlag, Berlin, 1981. [KW08] Alexander S. Kechris and W. Hugh Woodin. Generic codes for uncountable ordinals, partition properties, and elementary embeddings. In Alexander S. Kechris, Benedikt Löwe, and John R. Steel, editors, Games, Scales, and Suslin Cardinals, The Cabal Seminar, Volume I, volume 31 of Lecture Notes in Logic, pages 379–397. Cambridge University Press, 2008. [KLS08, pages 379–397]. [Sch99] Ralf-Dieter Schindler. Successive weakly compact or singular cardinals. The Journal of Symbolic Logic, 64(1):139–146, 1999. [Ste81a] John R. Steel. Closure properties of pointclasses. In Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis, editors, Proceedings of the Caltech-UCLA Logic Seminar, 1977–79, volume 839 of Lecture Notes in Mathematics, pages 147–163. Springer-Verlag, Berlin, 1981. [Ste81b] John R. Steel. Determinateness and the separation property. The Journal of Symbolic Logic, 46(1):41–44, 1981. [Ste83] John R. Steel. Scales in L(R). In Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis, editors, Proceedings of the Caltech-UCLA logic seminar held during the academic years 1979/1981, volume 1019 of Lecture Notes in Mathematics, pages 107–156. Springer-Verlag, Berlin, 1983. Reprinted in [KLS08, pages 130–175].

(A. W. Apter) Department of Mathematics, Baruch College, City University of New York, One Bernard Baruch Way, New York, NY 10010, United States of America; The CUNY Graduate Center, Mathematics, 365 Fifth Avenue, New York, NY 10016, United States of America E-mail address: [email protected] URL: http://faculty.baruch.cuny.edu/apter

(S. C. Jackson) Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, TX 76203-1430, United States of America E-mail address: [email protected] URL: http://www.math.unt.edu/~sjackson

(B. L¨owe) Institute for Logic, Language and Computation, Universiteit van Ams- terdam, Postbus 94242, 1090 GE Amsterdam, The Netherlands; Department Mathe- matik, Universitat¨ Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany; Mathematis- ches Institut, Rheinische Friedrich-Wilhelms-Universitat¨ Bonn, Endenicher Allee 60, 53115 Bonn, Germany E-mail address: [email protected] URL: http://staff.science.uva.nl/~bloewe